Lesson 15
A STORY OF RATIOS
8•2
Lesson 15: Informal Proof of the Pythagorean Theorem
Classwork
Example 1
Now that we know what the Pythagorean theorem is, let’s practice using it to find the length of a hypotenuse of a right
triangle.
Determine the length of the hypotenuse of the right triangle.
The Pythagorean theorem states that for right triangles 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 , where 𝑎𝑎 and 𝑏𝑏 are the legs and 𝑐𝑐 is the
hypotenuse. Then,
𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2
62 + 82 = 𝑐𝑐 2
36 + 64 = 𝑐𝑐 2
100 = 𝑐𝑐 2 .
Since we know that 100 = 102 , we can say that the hypotenuse 𝑐𝑐 = 10.
Example 2
Determine the length of the hypotenuse of the right triangle.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.83
Lesson 15
A STORY OF RATIOS
8•2
Exercises 1–5
For each of the exercises, determine the length of the hypotenuse of the right triangle shown. Note: Figures not drawn
to scale.
1.
2.
3.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.84
Lesson 15
A STORY OF RATIOS
8•2
4.
5.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.85
Lesson 15
A STORY OF RATIOS
8•2
Lesson Summary
Given a right triangle 𝐴𝐴𝐴𝐴𝐴𝐴 with 𝐶𝐶 being the vertex of the right angle, then the sides 𝐴𝐴𝐴𝐴 and 𝐵𝐵𝐵𝐵 are called the legs
of ∆𝐴𝐴𝐴𝐴𝐴𝐴, and 𝐴𝐴𝐴𝐴 is called the hypotenuse of ∆𝐴𝐴𝐴𝐴𝐴𝐴.
Take note of the fact that side 𝑎𝑎 is opposite the angle 𝐴𝐴, side 𝑏𝑏 is opposite the angle 𝐵𝐵, and side 𝑐𝑐 is opposite the
angle 𝐶𝐶.
The Pythagorean theorem states that for any right triangle, 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 .
Problem Set
For each of the problems below, determine the length of the hypotenuse of the right triangle shown. Note: Figures not
drawn to scale.
1.
2.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.86
Lesson 15
A STORY OF RATIOS
8•2
3.
4.
5.
6.
7.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.87
Lesson 15
A STORY OF RATIOS
8•2
8.
9.
10.
11.
12.
Lesson 15:
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Informal Proof of the Pythagorean Theorem
S.88
Lesson 16
A STORY OF RATIOS
8•2
Lesson 16: Applications of the Pythagorean Theorem
Classwork
Example 1
Given a right triangle with a hypotenuse with length 13 units and a leg with length 5 units, as shown, determine the
length of the other leg.
52 + 𝑏𝑏 2
5 − 52 + 𝑏𝑏 2
𝑏𝑏 2
𝑏𝑏 2
𝑏𝑏 2
𝑏𝑏
2
= 132
= 132 − 52
= 132 − 52
= 169 − 25
= 144
= 12
The length of the leg is 12 units.
Exercises 1–2
1.
Use the Pythagorean theorem to find the missing length of the leg in the right triangle.
Lesson 16:
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Applications of the Pythagorean Theorem
S.89
Lesson 16
A STORY OF RATIOS
2.
8•2
You have a 15-foot ladder and need to reach exactly 9 feet up the wall. How far away from the wall should you
place the ladder so that you can reach your desired location?
Exercises 3–6
3.
Find the length of the segment 𝐴𝐴𝐴𝐴, if possible.
Lesson 16:
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Applications of the Pythagorean Theorem
S.90
Lesson 16
A STORY OF RATIOS
8•2
4.
Given a rectangle with dimensions 5 cm and 10 cm, as shown, find the length of the diagonal, if possible.
5.
A right triangle has a hypotenuse of length 13 in. and a leg with length 4 in. What is the length of the other leg?
6.
Find the length of 𝑏𝑏 in the right triangle below, if possible.
Lesson 16:
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Applications of the Pythagorean Theorem
S.91
Lesson 16
A STORY OF RATIOS
8•2
Lesson Summary
The Pythagorean theorem can be used to find the unknown length of a leg of a right triangle.
An application of the Pythagorean theorem allows you to calculate the length of a diagonal of a rectangle, the
distance between two points on the coordinate plane, and the height that a ladder can reach as it leans against a
wall.
Problem Set
1.
Find the length of the segment 𝐴𝐴𝐴𝐴 shown below, if possible.
2.
A 20-foot ladder is placed 12 feet from the wall, as shown. How high up the wall will the ladder reach?
Lesson 16:
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Applications of the Pythagorean Theorem
S.92
Lesson 16
A STORY OF RATIOS
3.
8•2
A rectangle has dimensions 6 in. by 12 in. What is the length of the diagonal of the rectangle?
Use the Pythagorean theorem to find the missing side lengths for the triangles shown in Problems 4–8.
4.
Determine the length of the missing side, if possible.
5.
Determine the length of the missing side, if possible.
6.
Determine the length of the missing side, if possible.
Lesson 16:
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Applications of the Pythagorean Theorem
S.93
Lesson 16
A STORY OF RATIOS
7.
Determine the length of the missing side, if possible.
8.
Determine the length of the missing side, if possible.
Lesson 16:
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Applications of the Pythagorean Theorem
8•2
S.94
Lesson 13
A STORY OF RATIOS
8•3
Lesson 13: Proof of the Pythagorean Theorem
Classwork
Exercises 1–3
Use the Pythagorean Theorem to determine the unknown length of the right triangle.
1.
Determine the length of side 𝑐𝑐 in each of the triangles below.
a.
b.
2.
Determine the length of side 𝑏𝑏 in each of the triangles below.
a.
b.
Lesson 13:
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Proof of the Pythagorean Theorem
S.68
Lesson 13
A STORY OF RATIOS
3.
8•3
Determine the length of 𝑄𝑄𝑄𝑄. (Hint: Use the Pythagorean Theorem twice.)
Lesson 13:
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Proof of the Pythagorean Theorem
S.69
Lesson 13
A STORY OF RATIOS
8•3
Problem Set
Use the Pythagorean Theorem to determine the unknown length of the right triangle.
1.
Determine the length of side 𝑐𝑐 in each of the triangles below.
a.
b.
2.
Determine the length of side 𝑎𝑎 in each of the triangles below.
a.
a.
b.
Lesson 13:
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Proof of the Pythagorean Theorem
S.70
Lesson 13
A STORY OF RATIOS
3.
8•3
Determine the length of side 𝑏𝑏 in each of the triangles below.
a.
b.
4.
Determine the length of side 𝑎𝑎 in each of the triangles below.
a.
b.
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?
Lesson 13:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Proof of the Pythagorean Theorem
S.71
Lesson 14
A STORY OF RATIOS
8•3
Lesson 14: The Converse of the Pythagorean Theorem
Classwork
Exercises 1–7
1.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
2.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
3.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
Lesson 14:
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The Converse of the Pythagorean Theorem
S.72
Lesson 14
A STORY OF RATIOS
8•3
4.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
5.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
6.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
7.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
Lesson 14:
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The Converse of the Pythagorean Theorem
S.73
Lesson 14
A STORY OF RATIOS
8•3
Lesson Summary
The converse of the Pythagorean Theorem states that if side lengths of a triangle 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, satisfy 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 , then
the triangle is a right triangle.
If the side lengths of a triangle 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, do not satisfy 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 , then the triangle is not a right triangle.
Problem Set
1.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
2.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
3.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
Lesson 14:
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The Converse of the Pythagorean Theorem
S.74
Lesson 14
A STORY OF RATIOS
8•3
4.
The numbers in the diagram below indicate the units of length of each side of the triangle. Sam said that the
following triangle is a right triangle. Explain to Sam what he did wrong to reach this conclusion and what the correct
solution is.
5.
The numbers in the diagram below indicate the units of length of each side of the triangle. Is the triangle shown
below a right triangle? Show your work, and answer in a complete sentence.
6.
Jocelyn said that the triangle below is not a right triangle. Her work is shown below. Explain what she did wrong,
and show Jocelyn the correct solution.
We need to check if 272 + 452 = 362 is a true statement. The left side of the equation is equal to 2,754. The right
side of the equation is equal to 1,296. That means 272 + 452 = 362 is not true, and the triangle shown is not a
right triangle.
Lesson 14:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
The Converse of the Pythagorean Theorem
S.75
Lesson 1
A STORY OF RATIOS
8•7
Lesson 1: The Pythagorean Theorem
Classwork
Note: The figures in this lesson are not drawn to scale.
Example 1
Write an equation that will allow you to determine the length of the unknown side of the right triangle.
Example 2
Write an equation that will allow you to determine the length of the unknown side of the right triangle.
Lesson 1:
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The Pythagorean Theorem
1
Lesson 1
A STORY OF RATIOS
8•7
Example 3
Write an equation to determine the length of the unknown side of the right triangle.
Example 4
In the figure below, we have an equilateral triangle with a height of 10 inches. What do we know about an equilateral
triangle?
Exercises
1.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right
triangle. Explain why your estimate makes sense.
Lesson 1:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
The Pythagorean Theorem
2
Lesson 1
A STORY OF RATIOS
2.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right
triangle. Explain why your estimate makes sense.
3.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right
triangle. Explain why your estimate makes sense.
Lesson 1:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
The Pythagorean Theorem
8•7
3
Lesson 1
A STORY OF RATIOS
8•7
Lesson Summary
Perfect square numbers are those that are a product of an integer factor multiplied by itself. For example, the
number 25 is a perfect square number because it is the product of 5 multiplied by 5.
When the square of the length of an unknown side of a right triangle is not equal to a perfect square, you can
estimate the length as a whole number by determining which two perfect squares the square of the length is
between.
Example:
Let 𝑐𝑐 represent the length of the hypotenuse. Then,
32 + 72 = 𝑐𝑐 2
9 + 49 = 𝑐𝑐 2
58 = 𝑐𝑐 2 .
The number 58 is not a perfect square, but it is between the perfect squares 49 and 64.
Therefore, the length of the hypotenuse is between 7 and 8 but closer to 8 because 58 is
closer to the perfect square 64 than it is to the perfect square 49.
Problem Set
1.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
Lesson 1:
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The Pythagorean Theorem
4
Lesson 1
A STORY OF RATIOS
8•7
2.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
3.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
4.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
Lesson 1:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
The Pythagorean Theorem
5
Lesson 1
A STORY OF RATIOS
8•7
5.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
6.
Determine the length of the unknown side of the right triangle. Explain how you know your answer is correct.
7.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your
estimate makes sense.
Lesson 1:
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The Pythagorean Theorem
6
Lesson 1
A STORY OF RATIOS
8•7
8.
The triangle below is an isosceles triangle. Use what you know about the Pythagorean theorem to determine the
approximate length of the base of the isosceles triangle.
9.
Give an estimate for the area of the triangle shown below. Explain why it is a good estimate.
Lesson 1:
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The Pythagorean Theorem
7
Lesson 15
A STORY OF RATIOS
8•7
Lesson 15: Pythagorean Theorem, Revisited
Classwork
Proof of the Pythagorean Theorem
Lesson 15:
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Pythagorean Theorem, Revisited
76
Lesson 15
A STORY OF RATIOS
8•7
Discussion
Lesson 15:
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Pythagorean Theorem, Revisited
77
Lesson 15
A STORY OF RATIOS
8•7
Lesson Summary
The Pythagorean theorem can be proven by showing that the sum of the areas of the squares constructed off of the
legs of a right triangle is equal to the area of the square constructed off of the hypotenuse of the right triangle.
Problem Set
1.
For the right triangle shown below, identify and use similar triangles to illustrate the Pythagorean theorem.
2.
For the right triangle shown below, identify and use squares formed by the sides of the triangle to illustrate the
Pythagorean theorem.
Lesson 15:
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Pythagorean Theorem, Revisited
78
Lesson 15
A STORY OF RATIOS
8•7
3.
Reese claimed that any figure can be drawn off the sides
of a right triangle and that as long as they are similar
figures, then the sum of the areas off of the legs will equal
the area off of the hypotenuse. She drew the diagram by
constructing rectangles off of each side of a known right
triangle. Is Reese’s claim correct for this example? In
order to prove or disprove Reese’s claim, you must first
show that the rectangles are similar. If they are, then you
can use computations to show that the sum of the areas
of the figures off of the sides 𝑎𝑎 and 𝑏𝑏 equals the area of
the figure off of side 𝑐𝑐.
4.
After learning the proof of the Pythagorean theorem using
areas of squares, Joseph got really excited and tried
explaining it to his younger brother. He realized during his
explanation that he had done something wrong. Help
Joseph find his error. Explain what he did wrong.
5.
Draw a right triangle with squares constructed off of each side that Joseph can use the next time he wants to show
his younger brother the proof of the Pythagorean theorem.
6.
Explain the meaning of the Pythagorean theorem in your own words.
7.
Draw a diagram that shows an example illustrating the Pythagorean theorem.
Lesson 15:
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Pythagorean Theorem, Revisited
79
Lesson 16
A STORY OF RATIOS
8•7
Lesson 16: Converse of the Pythagorean Theorem
Classwork
Proof of the Converse of the Pythagorean Theorem
Exercises 1–7
1.
Is the triangle with leg lengths of 3 mi., 8 mi., and hypotenuse of length √73 mi. a right triangle? Show your work,
and answer in a complete sentence.
Lesson 16:
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Converse of the Pythagorean Theorem
80
Lesson 16
A STORY OF RATIOS
8•7
2.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
3.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
4.
Is the triangle with leg lengths of 9 in., 9 in., and hypotenuse of length √175 in. a right triangle? Show your work,
and answer in a complete sentence.
5.
Is the triangle with leg lengths of √28 cm, 6 cm, and hypotenuse of length 8 cm a right triangle? Show your work,
and answer in a complete sentence.
Lesson 16:
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Converse of the Pythagorean Theorem
81
Lesson 16
A STORY OF RATIOS
6.
8•7
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence.
.
7.
The triangle shown below is an isosceles right triangle. Determine the length of the legs of the triangle. Show your
work, and answer in a complete sentence.
Lesson 16:
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Converse of the Pythagorean Theorem
82
Lesson 16
A STORY OF RATIOS
8•7
Lesson Summary
The converse of the Pythagorean theorem states that if a triangle with side lengths 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 satisfies
𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 , then the triangle is a right triangle.
The converse can be proven using concepts related to congruence.
Problem Set
1.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
2.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
3.
Is the triangle with leg lengths of √3 cm, 9 cm, and hypotenuse of length √84 cm a right triangle? Show your work,
and answer in a complete sentence.
4.
Is the triangle with leg lengths of √7 km, 5 km, and hypotenuse of length √48 km a right triangle? Show your work,
and answer in a complete sentence.
Lesson 16:
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Converse of the Pythagorean Theorem
83
Lesson 16
A STORY OF RATIOS
8•7
5.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
6.
Is the triangle with leg lengths of 3, 6, and hypotenuse of length √45 a right triangle? Show your work, and answer
in a complete sentence.
7.
What is the length of the unknown side of the right triangle shown below? Show your work, and answer in a
complete sentence. Provide an exact answer and an approximate answer rounded to the tenths place.
8.
Is the triangle with leg lengths of 1 and √3 and hypotenuse of length 2 a right triangle? Show your work, and
answer in a complete sentence.
9.
Corey found the hypotenuse of a right triangle with leg lengths of 2 and 3 to be √13. Corey claims that since
√13 = 3.61 when estimating to two decimal digits, that a triangle with leg lengths of 2 and 3 and a hypotenuse of
3.61 is a right triangle. Is he correct? Explain.
10. Explain a proof of the Pythagorean theorem.
11. Explain a proof of the converse of the Pythagorean theorem.
Lesson 16:
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Converse of the Pythagorean Theorem
84
Lesson 17
A STORY OF RATIOS
8•7
Lesson 17: Distance on the Coordinate Plane
Classwork
Example 1
What is the distance between the two points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane?
What is the distance between the two points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane?
Lesson 17:
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Distance on the Coordinate Plane
85
Lesson 17
A STORY OF RATIOS
8•7
What is the distance between the two points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane? Round your answer to the tenths place.
Example 2
Given two points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane, determine the distance between them. First, make an estimate;
then, try to find a more precise answer. Round your answer to the tenths place.
Lesson 17:
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Distance on the Coordinate Plane
86
Lesson 17
A STORY OF RATIOS
8•7
Exercises 1–4
For each of the Exercises 1–4, determine the distance between points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane. Round your
answer to the tenths place.
1.
2.
Lesson 17:
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Distance on the Coordinate Plane
87
Lesson 17
A STORY OF RATIOS
8•7
3.
4.
Lesson 17:
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Distance on the Coordinate Plane
88
Lesson 17
A STORY OF RATIOS
8•7
Example 3
Is the triangle formed by the points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶 a right triangle?
Lesson 17:
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Distance on the Coordinate Plane
89
Lesson 17
A STORY OF RATIOS
8•7
Lesson Summary
To determine the distance between two points on the coordinate plane, begin by connecting the two points. Then,
draw a vertical line through one of the points and a horizontal line through the other point. The intersection of the
vertical and horizontal lines forms a right triangle to which the Pythagorean theorem can be applied.
To verify if a triangle is a right triangle, use the converse of the Pythagorean theorem.
Problem Set
For each of the Problems 1–4, determine the distance between points 𝐴𝐴 and 𝐵𝐵 on the coordinate plane. Round your
answer to the tenths place.
1.
2.
Lesson 17:
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Distance on the Coordinate Plane
90
Lesson 17
A STORY OF RATIOS
8•7
3.
4.
Lesson 17:
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Distance on the Coordinate Plane
91
Lesson 17
A STORY OF RATIOS
5.
8•7
Is the triangle formed by points 𝐴𝐴, 𝐵𝐵, 𝐶𝐶 a right triangle?
Lesson 17:
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Distance on the Coordinate Plane
92
Lesson 18
A STORY OF RATIOS
8•7
Lesson 18: Applications of the Pythagorean Theorem
Classwork
Exercises
1.
The area of the right triangle shown below is 26.46 in2 . What is the perimeter of the right triangle? Round your
answer to the tenths place.
Lesson18:
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Applications of the Pythagorean Theorem
93
Lesson 18
A STORY OF RATIOS
2.
8•7
The diagram below is a representation of a soccer goal.
a.
Determine the length of the bar, 𝑐𝑐, that would be needed to provide structure to the goal. Round your answer
to the tenths place.
b.
How much netting (in square feet) is needed to cover the entire goal?
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3.
8•7
The typical ratio of length to width that is used to produce televisions is 4: 3.
a.
A TV with those exact measurements would be quite small, so generally the size of the television is enlarged by
multiplying each number in the ratio by some factor of 𝑥𝑥. For example, a reasonably sized television might
have dimensions of 4 × 5: 3 × 5, where the original ratio 4: 3 was enlarged by a scale factor of 5. The size of a
television is described in inches, such as a 60" TV, for example. That measurement actually refers to the
diagonal length of the TV (distance from an upper corner to the opposite lower corner). What measurement
would be applied to a television that was produced using the ratio of 4 × 5: 3 × 5?
b.
A 42" TV was just given to your family. What are the length and width measurements of the TV?
c.
Check that the dimensions you got in part (b) are correct using the Pythagorean theorem.
d.
The table that your TV currently rests on is 30" in length. Will the new TV fit on the table? Explain.
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4.
5.
8•7
Determine the distance between the following pairs of points. Round your answer to the tenths place. Use graph
paper if necessary.
a.
(7, 4) and (−3, −2)
b.
(−5, 2) and (3, 6)
c.
Challenge: (𝑥𝑥1 , 𝑦𝑦1 ) and (𝑥𝑥2 , 𝑦𝑦2 ). Explain your answer.
What length of ladder will be needed to reach a height of 7 feet along the wall when the base of the ladder is 4 feet
from the wall? Round your answer to the tenths place.
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8•7
Problem Set
1.
2.
3.
A 70" TV is advertised on sale at a local store. What are the length and width of the television?
There are two paths that one can use to go from Sarah’s house to James’s house. One way is to take C Street, and
the other way requires you to use A Street and B Street. How much shorter is the direct path along C Street?
An isosceles right triangle refers to a right triangle with equal leg lengths, 𝑠𝑠, as shown below.
What is the length of the hypotenuse of an isosceles right triangle with a leg length of 9 cm? Write an exact answer
using a square root and an approximate answer rounded to the tenths place.
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4.
5.
6.
8•7
The area of the right triangle shown at right is 66.5 cm2 .
a.
What is the height of the triangle?
b.
What is the perimeter of the right triangle? Round your answer
to the tenths place.
What is the distance between points (1, 9) and (−4, −1)? Round your answer to the tenths place.
An equilateral triangle is shown below. Determine the area of the triangle. Round your answer to the tenths place.
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