lesson 3 homework practice the pythagorean theorem answer key

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Unit 9: Pythagorean theorem

About this unit, pythagorean theorem.

• Intro to the Pythagorean theorem (Opens a modal)
• Pythagorean theorem example (Opens a modal)
• Pythagorean theorem intro problems (Opens a modal)
• Use Pythagorean theorem to find right triangle side lengths 7 questions Practice
• Use Pythagorean theorem to find isosceles triangle side lengths 7 questions Practice
• Right triangle side lengths 4 questions Practice
• Use area of squares to visualize Pythagorean theorem 4 questions Practice

Pythagorean theorem application

• Pythagorean theorem word problem: carpet (Opens a modal)
• Pythagorean theorem word problem: fishing boat (Opens a modal)
• Pythagorean theorem in 3D (Opens a modal)
• Use Pythagorean theorem to find perimeter 4 questions Practice
• Pythagorean theorem word problems 4 questions Practice
• Pythagorean theorem in 3D 4 questions Practice
• Pythagorean theorem challenge 4 questions Practice

Pythagorean theorem and distance between points

• Distance formula (Opens a modal)
• Distance formula review (Opens a modal)
• Distance between two points 7 questions Practice

Pythagorean theorem proofs

• Garfield's proof of the Pythagorean theorem (Opens a modal)
• Bhaskara's proof of the Pythagorean theorem (Opens a modal)
• Pythagorean theorem proof using similarity (Opens a modal)
• Another Pythagorean theorem proof (Opens a modal)

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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• Grade 8 HMH Go Math - Answer Keys

Explanation:

When a coordinate grid is superimposed on a map of Harrisburg, the high school is located at (17, 21) and the town park is located at (28, 13). If each unit represents 1 mile, how many miles apart are the high school and the town park? Round your answer to the nearest tenth.

The coordinates of the vertices of a rectangle are given by R(- 3, - 4), E(- 3, 4), C (4, 4), and T (4, - 4). Plot these points on the coordinate plane at the right and connect them to draw the rectangle. Then connect points E and T to form diagonal \(\overline{ET}\).

a. Use the Pythagorean Theorem to find the exact length of \(\overline{ET}\).

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b. How can you use the Distance Formula to find the length of \(\overline{ET}\) ? Show that the Distance Formula gives the same answer.

Multistep The locations of three ships are represented on a coordinate grid by the following points: P(- 2, 5), Q(- 7, - 5), and R(2, - 3). Which ships are farthest apart?

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The Pythagorean Theorem Lesson Plan

Get the lesson materials.

The Pythagorean Theorem Guided Notes with Doodles Legs Hypotenuse Right Triangle

Ever wondered how to teach the Pythagorean Theorem in an engaging way to your 8th grade students?

In this lesson plan, students will learn about the Pythagorean Theorem proofs, legs and hypotenuse, right triangles, and their real-life applications. Through artistic, interactive guided notes, check for understanding, a color by code activity, and a real-life application example, students will gain a comprehensive understanding of the Pythagorean Theorem.

The lesson culminates with an exploration of baseball diamond highlighting the importance of understanding the Pythagorean Theorem in real-world scenarios.

• Type : Lesson Plans
• Grade : 8th Grade
• Standard : CCSS 8.G.B.7

Learning Objectives

After this lesson, students will be able to:

Identify the hypotenuse and legs of a right triangle

Use the Pythagorean Theorem to find the length of a missing side in a right triangle

Apply the Pythagorean Theorem to solve real-life problems

Prerequisites

Before this lesson, students should be familiar with:

Basic understanding of triangles (including how to identify right triangles) and their properties, such as side lengths and angle measures

Basic algebra skills, including solving for unknown variables and simplifying equations

Familiarity with squaring and square roots

Colored pencils or markers

Pythagorean Theorem

Guided Notes

Key Vocabulary

Right triangle

Square Roots

Introduction

As a hook, ask students why the Pythagorean Theorem is important in real life, such as finding the dimensions of a baseball diamond. Use the guided notes to introduce the Pythagorean Theorem.

Walk through the key points of the topic of the guided notes to teach. Refer to the FAQs for a walk-through on this, as well as ideas on how to respond to common student questions.

For Page 1 of the guided notes, introduce the Pythagorean Theorem with the worked example "what do you notice."

For Page 2 of the guided notes, explain how to know which side is a, b, or c, and check for understanding. Based on student responses, reteach concepts that students need extra help with.

Have students practice using the Pythagorean Theorem with the Maze activity. Walk around to answer student questions.

Fast finishers can dive into the Doodle Math activity for extra practice. You can assign it as homework for the remainder of the class.

Real-Life Application

Bring the class back together, and introduce the concept of using the Pythagorean Theorem to determine the dimensions of a baseball diamond. Explain that the Pythagorean Theorem can be used to find the distance between any two points, even if they are not directly next to each other. In the case of a baseball diamond, the distance from home plate to first base, home plate to second base, and so on can be found by using the Pythagorean Theorem. Refer to the FAQ for more ideas on how to teach it!

Hands-on project

A fun, no-prep way for students to practice is my Pythagorean Spiral project . Students will use a square to draw successive triangles and create a spiral. They will then measure the spiral and solve for the length of the hypotenuse for every triangle in their Pythagorean spiral. At the end of the activity, students can turn their Pythagorean spiral into an art project.

Additional self-checking digital practice

If you’re looking for digital practice for the Pythagorean theorem, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

There’s two version depending on the focus with your students:

Finding the hypotenuse and leg

. It’s available with fall , winter , Valentine’s Day , and St. Patrick’s Day themed images.

Challenge: Finding the distance between two points

. It’s available with Valentine’s Day and St. Patrick’s Day

themed images.

What is the Pythagorean theorem? Open

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In equation form, it can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

How do you use the Pythagorean theorem to find the length of a missing side in a right triangle? Open

To use the Pythagorean theorem to find the length of a missing side in a right triangle, follow these steps:

Identify the two sides of the right triangle that are known.

Assign one of the sides as "a" and the other as "b."

Identify the hypotenuse of the right triangle

, which is the side opposite the right angle. Assign the hypotenuse as "c."

Plug the known values into the Pythagorean theorem

, which states that a^2 + b^2 = c^2. Solve for the unknown side by isolating it on one side of the equation and taking the square root of both sides.

For example, if the length of side a is 3 and the length of side b is 4, and you want to find the length of the hypotenuse (c), you would use the equation 3^2 + 4^2 = c^2. Simplifying, you get 9 + 16 = c^2, or 25 = c^2. Taking the square root of both sides gives you c = 5. Therefore, the length of the hypotenuse (c) is 5.

What is a hypotenuse? Open

A hypotenuse is the longest side of a right triangle and is opposite the right angle. It is the side that connects the two legs of the triangle.

What are the legs of a right triangle? Open

The legs of a right triangle are the two sides that form the right angle. They are usually labeled as 'a' and 'b'.

How do you identify a right triangle? Open

To identify a right triangle, you need to look for a triangle with one angle that measures 90 degrees, also known as a right angle. This means that one side of the triangle will be perpendicular to another side, forming the right angle.

What are some real-life applications of the Pythagorean theorem? Open

The Pythagorean theorem has a wide range of applications in real life. Some examples include:

Architecture:

Architects use the Pythagorean theorem to make sure that building corners are square, and to calculate the length of sloping roofs.

Navigation:

The Pythagorean theorem is used in navigation to determine distances between two points. This is particularly useful in aviation and marine navigation.

In sports, the Pythagorean theorem can be used to calculate the distance between bases on a baseball diamond, or to determine the length of a shot in basketball.

Surveyors use the Pythagorean theorem to measure distances between points on the earth's surface, and to calculate elevations.

In the classroom, students can apply the Pythagorean theorem to real-world problems, such as finding the distance between two points on a map or the height of a building. One particularly fun application is to use the Pythagorean theorem to find the dimensions of a baseball diamond, which involves using the theorem to calculate the distance between bases.

How do you find the dimensions of a baseball diamond using the Pythagorean theorem? Open

To find the dimensions of a baseball diamond using the Pythagorean theorem, you need to calculate the distance between the bases. Here's how:

Identify the two bases you want to calculate the distance between.

Draw a right triangle connecting the two bases and home plate, with the two bases forming the legs and the distance between them forming the hypotenuse.

Measure the distance between the bases and record it as one leg of the triangle.

Measure the distance from the starting base to home plate and record it as the other leg of the triangle.

Use the Pythagorean theorem (a^2 + b^2 = c^2) to solve for the hypotenuse, which represents the distance between the two bases.

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Eureka Math Grade 8 Module 3 Lesson 13 Answer Key

Engage ny eureka math 8th grade module 3 lesson 13 answer key, eureka math grade 8 module 3 lesson 13 exercise answer key.

Exercises Use the Pythagorean theorem to determine the unknown length of the right triangle.

Exercise 1. Determine the length of side c in each of the triangles below.

15 2 +|TS| 2 =25 2 225+|TS| 2 =625 225-225+|TS| 2 =625-225 |TS| 2 =400 |TS|=20 Since |QT|+|TS|=|QS|, then the length of side \(\overline{Q S}\) is 8+20, which is 28.

Eureka Math Grade 8 Module 3 Lesson 13 Exit Ticket Answer Key

Then, determine the length of side \(\overline{C D}\). 12 2 +CD 2 =13 2 144+CD 2 =169 CD 2 =169-144 CD 2 =25 CD=5 Adding the lengths of sides \(\overline{B C}\) and \(\overline{C D}\) determines the length of side \(\overline{B D}\); therefore, 5+9=14. \(\overline{B D}\) has a length of 14.

Eureka Math Grade 8 Module 3 Lesson 13 Problem Set Answer Key

Students practice using the Pythagorean theorem to find unknown lengths of right triangles.

Use the Pythagorean theorem to determine the unknown length of the right triangle.

Question 5. What did you notice in each of the pairs of Problems 1–4? How might what you noticed be helpful in solving problems like these? Answer: In each pair of problems, the problems and solutions were similar. For example, in Problem 1, part (a) showed the sides of the triangle were 6, 8, and 10, and in part (b), they were 0.6, 0.8, and 1. The side lengths in part (b) were a tenth of the value of the lengths in part (a). The same could be said about parts (a) and (b) of Problems 2–4. This might be helpful for solving problems in the future. If I am given side lengths that are decimals, then I could multiply them by a factor of 10 to make whole numbers, which are easier to work with. Also, if I know common numbers that satisfy the Pythagorean theorem, like side lengths of 3, 4, and 5, then I recognize them more easily in their decimal forms, that is, 0.3, 0.4, and 0.5.

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