unit 9 lesson 3 homework (applying the pythagorean theorem)

This is a hand-picked list of online activities, tutorials, and worksheets concerning the The Pythagorean Theorem.

Investigate the areas of the squares on the sides of right angled triangles using this interactive figure.

Watch a dynamic, geometric "proof without words" of the Pythagorean Theorem. Can you explain the proof?

Solve two puzzles that illustrate the proof of the Pythagorean Theorem.

. Three computer activities give students the opportunity to observe triangles, learn and use the Pythagorean Theorem and practice different ways of determining areas of triangles.

Free PDF worksheets where you can practice finding the unknown length of the hypotenuse of a right triangle or find the length of a missing leg!

Lots of free worksheets for high school geometry, including topics such as the Pythagorean Theorem, perimeter, area, volume, angle relationships, triangle theorems, similarity and congruence, logic, proofs, trig, polygons, and circles.

of this book and of its supplement , latter by David Chandler.

An inexpensive companion to any high school geometry course with excellent explanations. .

Here is a non-intimidating way to prepare students for formal geometry. workbooks introduce students to a wide range of geometric discoveries as they do step-by-step constructions. Using only a pencil, compass, and straightedge, students begin by drawing lines, bisecting angles, and reproducing segments. Later they do sophisticated constructions involving over a dozen steps-and are prompted to form their own generalizations. When they finish, students will have been introduced to 134 geometric terms and will be ready to tackle formal proofs.

Pythagorean Theorem

The pythagorean theorem.

If we have a right triangle, and we construct squares using the edges or sides of the right triangle (gray triangle in the middle), the area of the largest square built on the hypotenuse (the longest side) is equal to the sum of the areas of the squares built on the other two sides. This is the Pythagorean Theorem in a nutshell. By the way, this is also known as the Pythagoras’ Theorem .

Notice that we square (raised to the second power) the variables [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] to indicate areas. The sum of the smaller squares (orange and yellow) is equal to the largest square (blue).

The Pythagorean Theorem relates the three sides in a right triangle. To be specific, relating the two legs and the hypotenuse, the longest side.

The Pythagorean Theorem can be summarized in a short and compact equation as shown below.

Definition of Pythagorean Theorem

For a given right triangle, it states that the square of the hypotenuse, [latex]c[/latex], is equal to the sum of the squares of the legs, [latex]a[/latex] and [latex]b[/latex]. That is, [latex]{a^2} + {b^2} = {c^2}[/latex].

For a more general definition, we have:

In right a triangle, the square of longest side known as the hypotenuse is equal to the sum of the squares of the other two sides.

The Pythagorean Theorem guarantees that if we know the lengths of two sides of a right triangle, we can always determine the length of the third side.

Here are the three variations of the Pythagorean Theorem formulas:

Let’s go over some examples!

Examples of Applying the Pythagorean Theorem

Example 1: Find the length of the hypotenuse.

Our goal is to solve for the length of the hypotenuse. We are given the lengths of the two legs. We know two sides out of the three! This is enough information for the formula to work.

For the legs, it doesn’t matter which one we assign for [latex]a[/latex] or [latex]b[/latex]. The result will be the same. So if we let [latex]a=5[/latex], then [latex]b=7[/latex]. Substituting these values into the Pythagorean Formula equation, we get

To isolate the variable [latex]c[/latex], we take the square roots of both sides of the equation. That eliminates the square (power of 2) on the right side. And on the left, we simply have a square root of a number which is no big deal.

However, we need to be mindful here when we take the square root of a number. We want to consider only the principal square root or the positive square root since we are dealing with length. It doesn’t make any sense to have a negative length, thus we disregard the negative length!

Therefore, the length of the hypotenuse is [latex]\sqrt {74}[/latex] inches. If we wish to approximate it to the nearest tenth, we have [latex]8.6[/latex] inches.

Example 2: Find the length of the leg.

Just by looking at the figure above, we know that we have enough information to solve for the missing side. The reason is the measure of the two sides are given and the other leg is left as unknown. That’s two sides given out of the possible three.

Here, we can let [latex]a[/latex] or [latex]b[/latex] equal [latex]7[/latex]. It really doesn’t matter. So, for this, we let [latex]a=7[/latex]. That means we are solving for the leg [latex]b[/latex]. But for the hypotenuse, there’s no room for error. We have to be certain that we are assigning [latex]c[/latex] for the length, that is, for the longest side. In this case, the longest side has a measure of [latex]9[/latex] cm and that is the value we will assign for [latex]c[/latex], therefore [latex]c=9[/latex].

Let’s calculate the length of leg [latex]b[/latex]. We have [latex]a=7[/latex] and [latex]c=9[/latex].

Therefore, the length of the missing leg is [latex]4\sqrt 2[/latex] cm. Rounding it to two decimal places, we have [latex]5.66[/latex] cm.

Example 3: Do the sides [latex]17[/latex], [latex]15[/latex] and [latex]8[/latex] form a right triangle? If so, which sides are the legs and the hypotenuse?

If these are the sides of a right triangle then it must satisfy the Pythagorean Theorem. The sum of the squares of the shorter sides must be equal to the square to the longest side. Obviously, the sides [latex]8[/latex] and [latex]15[/latex] are shorter than [latex]17[/latex] so we will assume that they are the legs and [latex]17[/latex] is the hypotenuse. So we let [latex]a=8[/latex], [latex]b=15[/latex], and [latex]c=17[/latex].

Let’s plug these values into the Pythagorean equation and check if the equation is true.

Since we have a true statement, then we have a case of a right triangle! We can now say for sure that the shorter sides [latex]8[/latex] and [latex]15[/latex] are the legs of the right triangle while the longest side [latex]17[/latex] is the hypotenuse.

Example 4: A rectangle has a length of [latex]8[/latex] meters and a width of [latex]6[/latex] meters. What is the length of the diagonal of the rectangle?

The diagonal of a rectangle is just the line segment that connects two non-adjacent vertices. In the figure below, it is obvious that the diagonal is the hypotenuse of the right triangle while the two other sides are the legs which are [latex]8[/latex] and [latex]6[/latex].

If we let [latex]a=6[/latex] and [latex]b=8[/latex], we can solve for [latex]c[/latex] in the Pythagorean equation which is just the diagonal.

Therefore, the measure of the diagonal is [latex]10[/latex] meters.

Example 5: A ladder is leaning against a wall. The distance from the top of the ladder to the ground is [latex]20[/latex] feet. If the base of the ladder is [latex]4[/latex] feet away from the wall, how long is the ladder?

If you study the illustration, the length of the ladder is just the hypotenuse of the right triangle with legs [latex]20[/latex] feet and [latex]4[/latex] feet.

Again, we just need to perform direct substitution into the Pythagorean Theorem formula using the known values then solve for [latex]c[/latex] or the hypotenuse.

Therefore, the length of the ladder is [latex]4\sqrt {26}[/latex] feet or approximately [latex]20.4[/latex] feet.

Example 6: In a right isosceles triangle, the hypotenuse measures [latex]12[/latex] feet. What is the length of each leg?

Remember that a right isosceles triangle is a triangle that contains a 90-degree angle and two of its sides are congruent.

In the figure below, the hypotenuse is [latex]12[/latex] feet. The two legs are both labeled as [latex]x[/latex] since they are congruent.

Let’s substitute these values into the formula then solve for the value of [latex]x[/latex]. We know that [latex]x[/latex] is just the leg of the right isosceles triangle which is the unknown that we are trying to solve for.

Therefore, the leg of the right isosceles triangle is [latex]6\sqrt 2[/latex] feet. If we want an approximate value, it is [latex]8.49[/latex] feet, rounded to the nearest hundredth.

Example 7: The diagonal of the square below is [latex]2\sqrt 2[/latex]. Find its area.

We know the area of the square is given by the formula [latex]A=s^2[/latex] where [latex]s[/latex] is the side of the square. So that means we need to find the side of the square given its diagonal. If we look closely, the diagonal is simply the hypotenuse of a right triangle. More importantly, the legs of the right triangle are also congruent.

Since the legs are congruent, we can let it equal to [latex]x[/latex].

Substitute these values into the Pythagorean Theorem formula then solve for [latex]x[/latex].

We calculated the length of the leg to be [latex]2[/latex] units. It is also the side of the square. So to find the area of the square, we use the formula

[latex]A = {s^2}[/latex]

That means, the area is

[latex]A = {s^2} = {\left( 2 \right)^2} = 4[/latex]

Therefore, the area of the square is [latex]4[/latex] square units.

You might also like these tutorials:

• Pythagorean Theorem Practice Problems with Answers
• Pythagorean Triples
• Generating Pythagorean Triples

9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem

Learning objectives.

By the end of this section, you will be able to:

• Use the properties of angles
• Use the properties of triangles

Use the Pythagorean Theorem

Be Prepared 9.7

Before you get started, take this readiness quiz.

Solve: x + 3 + 6 = 11 . x + 3 + 6 = 11 . If you missed this problem, review Example 8.6 .

Be Prepared 9.8

Solve: a 45 = 4 3 . a 45 = 4 3 . If you missed this problem, review Example 6.42 .

Be Prepared 9.9

Simplify: 36 + 64 . 36 + 64 . If you missed this problem, review Example 5.72 .

So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.

Use the Properties of Angles

Are you familiar with the phrase ‘do a 180 ’? 180 ’? It means to turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is 180 180 degrees. See Figure 9.5 .

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex . An angle is named by its vertex. In Figure 9.6 , ∠ A ∠ A is the angle with vertex at point A . A . The measure of ∠ A ∠ A is written m ∠ A . m ∠ A .

We measure angles in degrees, and use the symbol ° ° to represent degrees. We use the abbreviation m m for the measure of an angle. So if ∠ A ∠ A is 27° , 27° , we would write m ∠ A = 27 . m ∠ A = 27 .

If the sum of the measures of two angles is 180° , 180° , then they are called supplementary angles . In Figure 9.7 , each pair of angles is supplementary because their measures add to 180° . 180° . Each angle is the supplement of the other.

If the sum of the measures of two angles is 90° , 90° , then the angles are complementary angles . In Figure 9.8 , each pair of angles is complementary, because their measures add to 90° . 90° . Each angle is the complement of the other.

Supplementary and Complementary Angles

If the sum of the measures of two angles is 180° , 180° , then the angles are supplementary.

If ∠ A ∠ A and ∠ B ∠ B are supplementary, then m ∠ A + m ∠ B = 180°. m ∠ A + m ∠ B = 180°.

If the sum of the measures of two angles is 90° , 90° , then the angles are complementary.

If ∠ A ∠ A and ∠ B ∠ B are complementary, then m ∠ A + m ∠ B = 90°. m ∠ A + m ∠ B = 90°.

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

Use a Problem Solving Strategy for Geometry Applications.

• Step 1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for and choose a variable to represent it.
• Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem and make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

Example 9.16

An angle measures 40° . 40° . Find ⓐ its supplement, and ⓑ its complement.

Try It 9.31

An angle measures 25° . 25° . Find its: ⓐ supplement ⓑ complement.

Try It 9.32

An angle measures 77° . 77° . Find its: ⓐ supplement ⓑ complement.

Did you notice that the words complementary and supplementary are in alphabetical order just like 90 90 and 180 180 are in numerical order?

Example 9.17

Two angles are supplementary. The larger angle is 30° 30° more than the smaller angle. Find the measure of both angles.

Try It 9.33

Two angles are supplementary. The larger angle is 100° 100° more than the smaller angle. Find the measures of both angles.

Try It 9.34

Two angles are complementary. The larger angle is 40° 40° more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in Figure 9.9 is called Δ A B C , Δ A B C , read ‘triangle ABC ABC ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The three angles of a triangle are related in a special way. The sum of their measures is 180° . 180° .

Sum of the Measures of the Angles of a Triangle

For any Δ A B C , Δ A B C , the sum of the measures of the angles is 180° . 180° .

Example 9.18

The measures of two angles of a triangle are 55° 55° and 82° . 82° . Find the measure of the third angle.

Try It 9.35

The measures of two angles of a triangle are 31° 31° and 128° . 128° . Find the measure of the third angle.

Try It 9.36

A triangle has angles of 49° 49° and 75° . 75° . Find the measure of the third angle.

Right Triangles

Some triangles have special names. We will look first at the right triangle . A right triangle has one 90° 90° angle, which is often marked with the symbol shown in Figure 9.10 .

If we know that a triangle is a right triangle, we know that one angle measures 90° 90° so we only need the measure of one of the other angles in order to determine the measure of the third angle.

Example 9.19

One angle of a right triangle measures 28° . 28° . What is the measure of the third angle?

Try It 9.37

One angle of a right triangle measures 56° . 56° . What is the measure of the other angle?

Try It 9.38

One angle of a right triangle measures 45° . 45° . What is the measure of the other angle?

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 9.20

The measure of one angle of a right triangle is 20° 20° more than the measure of the smallest angle. Find the measures of all three angles.

Try It 9.39

The measure of one angle of a right triangle is 50° 50° more than the measure of the smallest angle. Find the measures of all three angles.

Try It 9.40

The measure of one angle of a right triangle is 30° 30° more than the measure of the smallest angle. Find the measures of all three angles.

Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures . One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles have the same measures.

The two triangles in Figure 9.11 are similar. Each side of Δ A B C Δ A B C is four times the length of the corresponding side of Δ X Y Z Δ X Y Z and their corresponding angles have equal measures.

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in Δ A B C : Δ A B C :

the length a can also be written B C the length b can also be written A C the length c can also be written A B the length a can also be written B C the length b can also be written A C the length c can also be written A B

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

Example 9.21

Δ A B C Δ A B C and Δ X Y Z Δ X Y Z are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Try It 9.41

Δ A B C Δ A B C is similar to Δ X Y Z . Δ X Y Z . Find a . a .

Try It 9.42

Δ A B C Δ A B C is similar to Δ X Y Z . Δ X Y Z . Find y . y .

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around 500 500 BCE.

Remember that a right triangle has a 90° 90° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the 90° 90° angle is called the hypotenuse , and the other two sides are called the legs . See Figure 9.12 .

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The Pythagorean Theorem

In any right triangle Δ A B C , Δ A B C ,

where c c is the length of the hypotenuse a a and b b are the lengths of the legs.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation m m and defined it in this way:

For example, we found that 25 25 is 5 5 because 5 2 = 25 . 5 2 = 25 .

We will use this definition of square roots to solve for the length of a side in a right triangle.

Example 9.22

Use the Pythagorean Theorem to find the length of the hypotenuse.

Try It 9.43

Try it 9.44, example 9.23.

Use the Pythagorean Theorem to find the length of the longer leg.

Try It 9.45

Use the Pythagorean Theorem to find the length of the leg.

Try It 9.46

Example 9.24.

Kelvin is building a gazebo and wants to brace each corner by placing a 10-inch 10-inch wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

Try It 9.47

John puts the base of a 13-ft 13-ft ladder 5 5 feet from the wall of his house. How far up the wall does the ladder reach?

Try It 9.48

Randy wants to attach a 17-ft 17-ft string of lights to the top of the 15-ft 15-ft mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

ACCESS ADDITIONAL ONLINE RESOURCES

• Animation: The Sum of the Interior Angles of a Triangle
• Similar Polygons
• Example: Determine the Length of the Hypotenuse of a Right Triangle

Section 9.3 Exercises

Practice makes perfect.

In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.

In the following exercises, use the properties of angles to solve.

Find the supplement of a 135° 135° angle.

Find the complement of a 38° 38° angle.

Find the complement of a 27.5° 27.5° angle.

Find the supplement of a 109.5° 109.5° angle.

Two angles are supplementary. The larger angle is 56° 56° more than the smaller angle. Find the measures of both angles.

Two angles are supplementary. The smaller angle is 36° 36° less than the larger angle. Find the measures of both angles.

Two angles are complementary. The smaller angle is 34° 34° less than the larger angle. Find the measures of both angles.

Two angles are complementary. The larger angle is 52° 52° more than the smaller angle. Find the measures of both angles.

In the following exercises, solve using properties of triangles.

The measures of two angles of a triangle are 26° 26° and 98° . 98° . Find the measure of the third angle.

The measures of two angles of a triangle are 61° 61° and 84° . 84° . Find the measure of the third angle.

The measures of two angles of a triangle are 105° 105° and 31° . 31° . Find the measure of the third angle.

The measures of two angles of a triangle are 47° 47° and 72° . 72° . Find the measure of the third angle.

One angle of a right triangle measures 33° . 33° . What is the measure of the other angle?

One angle of a right triangle measures 51° . 51° . What is the measure of the other angle?

One angle of a right triangle measures 22.5 ° . 22.5 ° . What is the measure of the other angle?

One angle of a right triangle measures 36.5 ° . 36.5 ° . What is the measure of the other angle?

The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.

The measure of the smallest angle of a right triangle is 20° 20° less than the measure of the other small angle. Find the measures of all three angles.

The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

The angles in a triangle are such that the measure of one angle is 20° 20° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, Δ A B C Δ A B C is similar to Δ X Y Z . Δ X Y Z . Find the length of the indicated side.

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is 270 270 miles.

Find the distance from Los Angeles to San Francisco.

Find the distance from San Francisco to Las Vegas.

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

A 13-foot 13-foot string of lights will be attached to the top of a 12-foot 12-foot pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is 12 12 feet high and 16 16 feet wide. How long should the banner be to fit the garage door?

Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of 10 10 feet. What will the length of the path be?

Brian borrowed a 20-foot 20-foot extension ladder to paint his house. If he sets the base of the ladder 6 6 feet from the house, how far up will the top of the ladder reach?

Everyday Math

Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is 30 30 feet wide and 35 35 feet tall at the highest point of the roof. If the dollhouse will be 2.5 2.5 feet wide, how tall will its highest point be?

Measurement A city engineer plans to build a footbridge across a lake from point X X to point Y , Y , as shown in the picture below. To find the length of the footbridge, she draws a right triangle XYZ , XYZ , with right angle at X . X . She measures the distance from X X to Z , 800 Z , 800 feet, and from Y Y to Z , 1,000 Z , 1,000 feet. How long will the bridge be?

Writing Exercises

Write three of the properties of triangles from this section and then explain each in your own words.

Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length 3 3 and 4 . 4 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Using the Pythagorean Theorem and Similarity

• Let’s explore right triangles with altitudes drawn to the hypotenuse.

13.1: Similar, Right?

Is triangle \(ADC\) similar to triangle \(CDB\) ? Explain or show your reasoning.

13.2: Tangled Triangles

Trace the 2 smaller triangles onto separate pieces of tracing paper.

• Turn your tracing paper and convince yourself all 3 triangles are similar.
• Write 3 similarity statements.
• Determine the scale factor for each pair of triangles.
• Determine the lengths of sides \(HG\) , \(GF\) , and \(HF\) .

13.3: More Tangled Triangles

• Convince yourself there are 3 similar triangles. Write a similarity statement for the 3 triangles.
• Write as many equations about proportional side lengths as you can.
• What do you notice about these equations?

Tyler says that since triangle \(ACD\) is similar to triangle \(ABC\) , the length of \(CB\) is 11.96. Noah says that since \(ABC\) is a right triangle, we can use the Pythagorean Theorem. So the length of \(CB\) is 12 exactly. Do you agree with either of them? Explain or show your reasoning.

When we draw an altitude from the hypotenuse of a right triangle, we get lots of similar triangles that can be used to find missing lengths. An altitude is a segment from one vertex of the triangle to the line containing the opposite side that is perpendicular to the opposite side. For right triangle \(PQR\)  we can draw the altitude \(PS\) . 

Why are triangles  \(PQR\) , \(SQP\) , and \(SPR\)  all similar to each other?

Triangles \(PQR\) and \(SQP\) are similar by the Angle-Angle Triangle Similarity Theorem because angle \(Q\) is in both triangles, and both triangles are right triangles, so angles \(RPQ\) and \(PSQ\) are congruent. Triangles \(PQR\) and \(SPR\) are similar by the Angle-Angle Triangle Similarity Theorem because angle \(R\) is in both triangles, and both triangles are right triangles, so angles \(RPQ\) and \(RSP\) are congruent. Because triangles \(SQP\) and \(SPR\) are both similar to triangle \(PQR\) , they are also similar to each other.

Since the triangles \(PQR\) , \(SQP\) , and \(SPR\) are all similar, corresponding angles are congruent and pairs of corresponding sides are scaled copies of each other, by the same scale factor. We can use the proportionality of pairs of corresponding side lengths to find missing side lengths. For example, suppose we need to find \(PS\) and know \(RS=3\) and \(QS=7\) . Since triangle \(SQP\) is similar to triangle \(SPR\) , we know \(\frac{RS}{PS}=\frac{PS}{QS}\) . So \(\frac{3}{PS}=\frac{PS}{7}\) and \(PS=\sqrt{21}\) . Or, suppose we need to find \(SQ\) and know \(PQ=5\) and \(RQ=12\) . Since triangle \(PQR\) is similar to triangle \(SQP\) , we know \(\frac{RQ}{PQ}=\frac{PQ}{SQ}\) . So \(\frac{12}{5}=\frac{5}{SQ}\) and \(SQ=\frac{25}{12}\) .

Glossary Entries

An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.

Pythagorean Theorem End of Unit Review | Practice Worksheet

Also included in

Description

This 22 question worksheet is a review of skills in the 8th grade Pythagorean Theorem core standards. These Pythagorean theorem questions are perfect to see if your students are ready for a unit test or state test!

For this review students should be able to:

• Find the missing lengths from right triangles
• Use Pythagorean Theorem to find lengths in 2d and 3d figures
• Use the converse of the Pythagorean Theorem to determine if a given triangle is right, acute, or obtuse
• Use the Pythagorean Theorem to find the distance between points on a coordinate plane
• Use the Pythagorean Theorem to find the distance between points WITHOUT a coordinate plane
• Problem scenarios with the above skills

This worksheet is great for subs, homework, or to review in class before the end of unit assessment. It covers all standards on Pythagorean Theorem.

All files come as ready to use PDFs. Questions too hard or too easy? Want to edit the files? All files come with links to Google Docs/Slides where you can edit the files for your students.

CLICK HERE to see the lessons in this unit.

CLICK HERE to see activities that go along with this unit.

CLICK HERE to see the assessments that go along with this unit.

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Maneuvering the Middle

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Pythagorean Theorem Unit 8th Grade TEKS

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An 8 day Pythagorean Theorem TEKS-Aligned complete unit including: the Pythagorean Theorem, the Pythagorean Theorem converse, Pythagorean Theorem application.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills.  You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards:   TEKS: 8.6C, 8.7C, 8.7D;  Looking for CCSS-Aligned Resources?  Grab the Pythagorean Theorem CCSS-Aligned Unit.  Please don’t purchase both as there is overlapping content.

Learning Focus:

• use the pythagorean theorem and its converse to solve problems
• use models of the pythagorean theorem
• determine the distance between points on a coordinate plane using the pythagorean theorem

What is included in the 8th grade teks pythagorean theorem unit?

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning.
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our  Pythagorean Theorem Activity Bundle  for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for more 8th Grade Math Material? Join our All Access Membership Community! You can reach your students without the “I still have to prep for tomorrow” stress, the constant overwhelm of teaching multiple preps, and the hamster wheel demands of creating your own teaching materials.

• Grade Level Curriculum
• Supplemental Digital Components
• Complete and Comprehensive Student Video Library 

Click here to learn more about All Access by Maneuvering the Middle®!

Licensing: This file is a license for ONE teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

Customer Service: If you have any questions, please feel free to reach out for assistance .  We aim to provide quality resources to help teachers and students alike, so please reach out if you have any questions or concerns. 

Maneuvering the Middle ® Terms of Use: Products by Maneuvering the Middle®, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online in an LMS such as Google Classroom, Canvas, or Schoology. Students should be the only ones able to access the resources.  It is a copyright violation to upload the files to school/district servers or shared Google Drives. See more information on our terms of use here . 

If you are interested in a personalized quote for campus and district licenses, please click here . 

©Maneuvering the Middle® LLC, 2012-present

This file is a license for one teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

Customer Service

We strive to provide quality products to help teachers and students alike, so contact us with any questions.

Maneuvering the Middle® Terms of Use

Products by Maneuvering the Middle, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online if they are behind a password-protected site.

Campus and district licensing is available please contact us for pricing.

©Maneuvering the Middle LLC, 2012-present

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This resource is often paired with:

Digital Math Activity Bundle 8th Grade

Pythagorean Theorem Activity Bundle 8th Grade