We included **HMH Into Math Grade 8 Answer Key PDF** **Module 11 The Pythagorean Theorem **to make students experts in learning maths.

## HMH Into Math Grade 8 Module 11 Answer Key The Pythagorean Theorem

**Try Your Angle**

A worker is making wooden triangles to use as obstacles for minigolf. The sides of the triangles can have any of the lengths shown. A triangle can be isosceles, but none of the triangles can be equilateral.

Name the side lengths of four different triangles to be used as obstacles. No two triangles should have the same set of lengths.

Triangle 1: __________, ___________, and ___________

Triangle 2: __________, ___________, and ___________

Triangle 3: __________, ___________, and ___________

Triangle 4: __________, ___________, and ___________

Answer:

The side lengths of four different triangles to be used as obstacles are Triangle 1: 2 ft, 2 ft, and 3ft, Triangle 2: 3 ft, 3ft, and 4 ft, Triangle 3: 4 ft, 4ft, and 5ft, Triangle 4: 5 ft, 5ft, and 2 ft,

Explanation:

Given a worker is making wooden triangles to use as obstacles for minigolf.

The sides of the triangles can have any of the lengths shown. A triangle can be isosceles,

but none of the triangles can be equilateral.

Naming the side lengths of four different triangles to be used as obstacles. No two triangles should have the same set of lengths. An isosceles triangle has two equal side lengths and two equal angles, the corners at which these sides meet the third side is symmetrical in shape,

Given side lengths as 1 ft, 2 ft, 3 ft, 4 ft, 5 ft and we have if the sum of two side lengths of a triangle is always greater than the third side, So the side lengths of four different triangles to be used as obstacles are Triangle 1: 2 ft, 2 ft, and 3ft, 2 ft + 2 ft > 3 ft, 4 ft > 3 ft, Triangle 2: 3 ft, 3ft, and 4 ft,

3 ft + 3 ft > 4 ft, 6 ft > 4 ft, Triangle 3: 4 ft, 4ft, and 5ft, 4 ft + 4 ft > 5 ft, 8 ft > 5 ft,

Triangle 4: 5 ft, 5ft, and 2 ft, 5 ft + 5 ft > 2 ft, 10 ft > 2 ft.

**Turn and Talk**

How do you know that each set of lengths can form a triangle?

Answer:

If the sum of two side lengths of a triangle is

always greater than the third side,

Explanation:

Triangle Inequality Theorem, which states that the sum of two side lengths of a triangle is always

greater than the third side. If this is true for all three combinations of added side lengths,

then we will have a triangle.

**Are You Ready?**

Complete these problems to review prior concepts and skills you will need for this module.

**Order of Operations**

**Determine the value of each expression.**

Question 1.

18 + 7^{2} – 24 ___________

Answer:

43,

Explanation:

Given to determine the value of expression 18 + 7^{2} – 24 = 18 + (7 X 7) – 24 = 18 + 49 – 24 = 43.

Question 2.

2(10 – 4)^{2} + 8 ____________

Answer:

80,

Explanation:

Given to determine the value of expression 2(10 – 4)^{2} + 8 = 2(6)^{2} + 8 = 2 X (6 X 6) + 8 =

2 X (36) + 8 = 72 + 8 = 80.

Question 3.

8^{2} + 5^{2} _____________

Answer:

89,

Explanation:

Given to determine the value of expression 8^{2} + 5^{2 }= (8 X 8) + (5 X 5) = 64 + 25 = 89.

Question 4.

20^{2} – 12^{2} ____________

Answer:

256,

Explanation:

Given to determine the value of expression 20^{2} – 12^{2 }= (20 X 20) – (12 X 12) =

400 – 144 = 256.

**Draw Shapes with Given Conditions**

**For Problems 5-6, state whether a triangle can be formed from the set of side lengths. Write yes or no.**

Question 5.

1 centimeter, 2 centimeters, and 4 centimeters ____________

Answer:

No,

Explanation:

We know that sum of two sides of triangle will be greater than or equal to the third side. Since, the sum of 1 centimeter, 2 centimeters is not greater than the third side4 centimeters ,therefore, a triangle can not be formed from the set of side lengths.

Question 6.

2 centimeters, 2 centimeters, and 3 centimeters _____________

Answer:

Yes,

Explanation:

We know that sum of two sides of triangle will be greater than or equal to the third side. Since, the sum of 2 centimeters, 2 centimeters is greater than the third side, therefore, a triangle can be formed from the set of side lengths.

Question 7.

Two sides of a triangle measure 6 inches and 8 inches. What is a possible length of the third side? Explain your reasoning.

Answer:

2 < c < 14,

Explanation:

For any triangle, the sum of any two sides must be greater than the third side.

Let’s say we have triangle ABC, where a = 6 and b = 8.

a + b > c (1), a + c > b (2), b + c > a (3), From (1) 6 + 8 > c, c < 14, From (2): 6 + c > 8, c > 2,

From (3): 8 + c > 6, c > -2, So the possible lengths of c are: 2 < c < 14.

**Use Roots to Solve Equations**

**For Problems 8-11, solve the equation.**

Question 8.

a^{2} = 100

Answer:

a = 10,

Explanation:

The root of a^{2} = 100 is a^{2} = 10 X 10, So a = 10.

Question 9.

c^{2} = 35

Answer:

c = 5.916 almost 6,

Explanation:

The root of c^{2} = 35 is c^{2} = 5.916 X 5.916, So c = 5.916.

Question 10.

b^{2} = 144

Answer:

b= 12,

Explanation:

The root of b^{2} = 144 is b^{2} = 14 X 14, So b = 14.

Question 11.

x^{2} = 225

Answer:

x = 15,

Explanation:

The root of x^{2} = 225 is x^{2} = 15 X 15, So a = 15.

Question 12.

A square park has an area of 8100 square meters.

A. Write an equation that can be used to determine the side length s, in meters, of the park.

Answer:

s^{2} = 8,100,

Explanation:

Given a square park has an area of 8100 square meters.

An equation that can be used to determine the side length s in meters, of the park is s^{2} = 8,100.

B. Solve your equation, and interpret the solution.

Answer:

Side length is 90 meters,

Explanation:

Squares have equal side lengths, so that is why the question is only asking for one side length.

The area of a square is calculated using the formula s^2, where s stands for a side of the square.

To find the length of one side of the park, We can take the given area (8,100 m^2) and square root it.

This would give us the length of one side of the square park.

Solving equation s^{2} = 8,100 and finding s as √8,100 = 90, therefore the length of one side of the

park is 90 meters.