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

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

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 ___________
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?
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.

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 + 72 – 24 ___________
43,

Explanation:
Given to determine the value of expression 18 + 72 – 24 = 18 + (7 X 7) – 24 = 18 + 49 – 24 = 43.

Question 2.
2(10 – 4)2 + 8 ____________
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.
82 + 52 _____________
89,

Explanation:
Given to determine the value of expression 82 + 52 = (8 X 8) + (5 X 5) = 64 + 25 = 89.

Question 4.
202 – 122 ____________
256,

Explanation:
Given to determine the value of expression 202 – 122 = (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 ____________
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 _____________
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.
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.
a2 = 100
a = 10,

Explanation:
The root of a2 = 100 is a2 = 10 X 10, So a = 10.

Question 9.
c2 = 35
c = 5.916 almost 6,

Explanation:
The root of c2 = 35 is c2 = 5.916 X 5.916, So c = 5.916.

Question 10.
b2 = 144
b= 12,

Explanation:
The root of b2 = 144 is b2 = 14 X 14, So b = 14.

Question 11.
x2 = 225
x = 15,

Explanation:
The root of x2 = 225 is x2 = 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.
s2 = 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 s2 = 8,100.

B. Solve your equation, and interpret the solution.
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 s2 = 8,100 and finding s as √8,100 = 90, therefore the length of one side of the
park is 90 meters.

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