Geometry Workbook — Chapter 8: The Pythagorean Theorem and Special Right Triangles

SOL G.TR.4 (a, b, c, f) · Companion to Textbook Chapter 8

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 110.

PAGE 1 — Chapter opener

Chapter 8 · The Pythagorean Theorem and Special Right Triangles

Standard G.TR.4 (a, b, c, f)

In this chapter you will:

• Find a hypotenuse or a leg, and report the answer in simplest radical form
• Use the converse to sort three lengths into right, acute, or obtuse
• Recognize Pythagorean Triples and use one as a shortcut
• Solve 45°-45°-90° triangles in both directions, rationalizing when the hypotenuse is given
• Solve 30°-60°-90° triangles the same way, always finding the short leg first

Words to know: hypotenuse · leg · Pythagorean Theorem · converse · right · acute · obtuse · Pythagorean Triple · simplest radical form · rationalize the denominator · short leg · long leg · altitude · diagonal

Conventions: name the hypotenuse first. Two legs given: add. Hypotenuse and a leg: subtract. $2\sqrt{13}$ is the answer; $7.21$ is that answer rounded. Rationalize — $\tfrac{10}{\sqrt{2}}$ is unfinished.

PAGE 2 — The theorem

8.1 a² + b² = c²

FIGURE: fig1-the-pythagorean-theorem.png (full width)

Fill in the blanks.

The hypotenuse is the side opposite the ____________ angle, and it is always the ____________ .

1. Which side is the hypotenuse, and how do you know? ____________________

2. Areas of the three squares: ______ , ______ , ______

3. The equation: ______ + ______ = ______

PAGE 3 — Why it is true

Chapter 7 Doing Chapter 8's Work

FIGURE: fig2-proof-by-similarity.png (full width)

Fill in the blanks.

The altitude creates two triangles, each similar to the whole by ______ .

Adding the two proportions gives ____________________ .

4. What two triangles does the altitude create, and why is each similar to the whole?

   ---

5. Altitude = ______ Segments = ______ and ______

PAGE 4 — Two shapes, one equation

Add or Subtract, and Then Simplify

FIGURE: fig3-leg-or-hypotenuse.png (full width)

FIGURE: fig4-exact-before-rounded.png (half width)

Fill in the blanks.

√52 is ____________ . 7.21 is ____________ . 2√13 is ____________ .

6. Which problem adds the squares, and which subtracts? ____________________

PAGE 5 — Practice · the theorem

Practice

7. Legs $9$, $12$. $c =$ ______

8. Legs $7$, $24$. $c =$ ______

9. Hypotenuse $26$, leg $10$. other leg = ______

10. Hypotenuse $41$, leg $9$. other leg = ______

11. Legs $2$, $3$. $c =$ ______

12. Legs $5$, $5$. $c =$ ______

13. Legs $6$, $10$. $c =$ ______

14. Hypotenuse $12$, leg $8$. other leg = ______

15. Legs $1.5$, $2$. $c =$ ______

16. Legs $8$, $15$. $c =$ ______

17. Hypotenuse $25$, leg $7$. other leg = ______

18. Legs $3$, $7$. $c =$ ______

PAGE 6 — Apply and reason · Lesson 8.1

Think It Through

16. Application. A $13$ ft ladder, foot $5$ ft from the wall. Height reached = ______ ft

17. Application. A $6$ ft by $8$ ft gate. Diagonal brace = ______ ft

18. Error analysis. Given legs $6$ and $10$, a student writes $c^2 = 10^2 - 6^2$. Error and correct answer:

    ---

19. Why does identifying the hypotenuse come before writing the equation?

    ---

PAGE 7 — The converse

8.2 Three Lengths In, One Word Out

FIGURE: fig5-the-converse-classifies.png (full width)

Fill in the blanks.

$a^2 + b^2 = c^2$ → ____________ . $a^2 + b^2 > c^2$ → ____________ . $a^2 + b^2 < c^2$ → ____________ .

23. Which is right? ______ Comparison: ____________

24. Which is acute? ______ Comparison: ____________

25. Which is obtuse? ______ Comparison: ____________

PAGE 8 — Two checks first

Sort, Then Compare

FIGURE: fig6-classification-board.png (full width)

26. What must be done to the three lengths before comparing? ______

27. What else must be checked before classifying at all? ____________________

28. State the classification rule in one sentence.

    ---

PAGE 9 — Practice · classifying

Practice

29. $9$, $40$, $41$: ______ + ______ = ______ vs ______ → ______

30. $5$, $6$, $8$: ______ vs ______ → ______

31. $7$, $8$, $10$: ______ vs ______ → ______

32. $12$, $16$, $20$: ______ vs ______ → ______

33. $4$, $5$, $10$: ______

34. $10$, $24$, $26$: ______ vs ______ → ______

35. $2$, $3$, $4$: ______ vs ______ → ______

36. $6$, $7$, $9$: ______ vs ______ → ______

37. $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$: ______ vs ______ → ______

38. $11$, $60$, $61$: ______ vs ______ → ______

39. $5$, $12$, $13$: ______

40. $5$, $12$, $12$: ______

41. $5$, $12$, $14$: ______

PAGE 10 — Apply and reason · Lesson 8.2

Think It Through

39. Application. A carpenter measures $6$ ft, $8$ ft, and $10.2$ ft across.

    Square? ______ Comparison: ______ vs ______ Angle is ______ than $90°$

40. Error analysis. For $6$, $8$, $10$ a student compares $8^2 + 10^2$ with $6^2$ and says acute. Error and correction:

    ---

41. Name the two checks that come before comparing. ____________________

PAGE 11 — Triples

8.3 Three Whole Numbers

FIGURE: fig7-the-common-triples.png (full width)

Fill in the blanks.

Five worth knowing: ______ , ______ , ______ , ______ , ______

45. What is a Pythagorean Triple? ____________________

46. Verify $8$, $15$, $17$: ______ + ______ = ______ = ______ ²

47. Why is every multiple of a triple also a triple? ____________________

PAGE 12 — A shortcut, not a rule

Recognize One, Save the Arithmetic

FIGURE: fig8-a-triple-saves-the-arithmetic.png (full width)

48. The long way to the hypotenuse of a $9$-$12$ right triangle: ____________________

49. The short way: ____________________

50. When is the short way unavailable? ____________________

PAGE 13 — Practice · triples

Practice

51. $6$, $8$, $10$ a triple? ______ From: ______

52. $5$, $12$, $13$ a triple? ______ Check: ____________

53. $4$, $5$, $6$ a triple? ______ Check: ____________

54. Legs $21$, $28$. Triple used ______ $c =$ ______

55. Legs $15$, $36$. Triple used ______ $c =$ ______

56. Hypotenuse $50$, leg $30$. other leg = ______

57. Hypotenuse $85$, leg $40$. other leg = ______

58. Legs $14$, $48$. $c =$ ______

59. Legs $27$, $36$. $c =$ ______

60. Hypotenuse $65$, leg $25$. other leg = ______

61. Legs $18$, $24$. $c =$ ______

62. Hypotenuse $39$, leg $15$. other leg = ______

63. $9$, $12$, $16$ a triple? ______ Check: ____________

PAGE 14 — Apply and reason · Lesson 8.3

Think It Through

61. Application. A screen $48$ in by $20$ in. Diagonal = ______ in Triple used: ______

62. Reasoning. Why does multiplying all three numbers of a triple by $k$ give another triple?

    ---

63. State what a Pythagorean Triple is. ____________________

PAGE 15 — Half a square

8.4 The 45°-45°-90°

FIGURE: fig9-45-45-90.png (full width)

Fill in the blanks.

leg : leg : hypotenuse = ______ : ______ : ______

67. Where does this triangle come from? ____________________

68. State the ratio. ____________________

69. $5^2 + 5^2 =$ ______ and $\sqrt{50} =$ ______

PAGE 16 — Both directions

Multiply One Way, Rationalize the Other

FIGURE: fig10-45-45-90-both-directions.png (full width)

Fill in the blanks.

Leg given → ____________ by √2. Hypotenuse given → ____________ by √2, then ____________ .

70. Leg $7$ → hypotenuse = ______

71. Hypotenuse $10$ → leg = ______ Rationalizing shown: ____________________

72. Why is $\tfrac{10}{\sqrt{2}}$ an unfinished answer? ____________________

PAGE 17 — Practice · 45-45-90

Practice

73. Leg $9$ → hypotenuse ______

74. Leg $12$ → hypotenuse ______

75. Leg $3\sqrt{2}$ → hypotenuse ______

76. Hypotenuse $8$ → leg ______

77. Hypotenuse $14$ → leg ______

78. Hypotenuse $6\sqrt{2}$ → leg ______

79. Hypotenuse $5$ → leg ______

80. Square with side $11$ → diagonal ______

81. Square with diagonal $16$ → side ______

82. Leg $6$ → hypotenuse ______

83. Hypotenuse $20$ → leg ______

84. Square with side $7$ → diagonal ______

PAGE 18 — Apply and reason · Lesson 8.4

Think It Through

82. Application. A square tile $10$ in on a side, cut along the diagonal.

    Cut = ______ in exactly ≈ ______ in

83. Error analysis. A student says the leg is half the hypotenuse. Counterexample from this lesson:

    ---

84. Reasoning. Why is the hypotenuse always longer than a leg here — without computing?

    ---

85. State the ratio and which side each number belongs to. ____________________

PAGE 19 — Half an equilateral triangle

8.5 The 30°-60°-90°

FIGURE: fig11-30-60-90.png (full width)

Fill in the blanks.

short leg : long leg : hypotenuse = ______ : ______ : ______

The short leg is opposite the ______ ° angle and is ______ the hypotenuse.

89. Where does this triangle come from? ____________________

90. State the ratio. ____________________

91. Which leg is opposite the $30°$ angle, and what is it called? ____________________

PAGE 20 — Short leg first, always

Get Back to the Short Leg

FIGURE: fig12-30-60-90-both-directions.png (full width)

92. Short leg $6$ → long leg ______ hypotenuse ______

93. Long leg $9$ → short leg ______ hypotenuse ______ Rationalizing shown: ____________________

PAGE 21 — Both triangles in context

Diagonals and Altitudes

FIGURE: fig13-special-triangles-in-context.png (full width)

Fill in the blanks.

A square's diagonal is always side × ______ . An equilateral triangle's altitude is always half the side × ______ .

94. Square's diagonal = ______ m Equilateral triangle's altitude = ______ ft

PAGE 22 — Practice · 30-60-90

Practice

95. Short leg $5$ → long leg ______ hypotenuse ______

96. Short leg $11$ → long leg ______ hypotenuse ______

97. Hypotenuse $18$ → short leg ______ long leg ______

98. Hypotenuse $26$ → short leg ______ long leg ______

99. Long leg $12$ → short leg ______ hypotenuse ______

100. Long leg $5\sqrt{3}$ → short leg ______ hypotenuse ______

101. Long leg $21$ → short leg ______ hypotenuse ______

102. Equilateral side $14$ → altitude ______

103. Equilateral side $6$ → altitude ______

104. Equilateral altitude $9\sqrt{3}$ → side ______

105. Short leg $8$ → long leg ______ hypotenuse ______

106. Hypotenuse $30$ → short leg ______ long leg ______

107. Long leg $6$ → short leg ______ hypotenuse ______

PAGE 23 — Apply and reason · Lesson 8.5

Think It Through

105. Application. An equilateral sign $4$ ft on each side. Height = ______ ft exactly ≈ ______ ft

106. Error analysis. Given a long leg of $10$, a student doubles it for a hypotenuse of $20$. Error and correction:

     ---

107. State the ratio and which side each number belongs to. ____________________

PAGE 24 — Work frames

Blank Frames

FIGURE: fig14-blank-right-triangle-frames.png (full width)

Mark the right angle and name the hypotenuse before writing anything else. Then decide: two legs given means add, a hypotenuse and a leg means subtract.

Reminder. Finish every answer. A radical is simplified when no square factor is left under it and no radical is left in a denominator.

PAGE 25 — Chapter review

Review

Review 1 (G.TR.4 a, f). Measurements of $9$ m, $12$ m, and $15$ m across.

• Square? ______ Theorem used: ____________ Comparison: ______ vs ______
• Triple ______ Multiplier ______
• Second bed $9$, $12$, $16$: ______ vs ______ → ______
• Third measurement $9$, $12$, $22$: why it cannot be classified: ____________________

Review 2 (G.TR.4 b, c). A square frame $18$ in on a side; an equilateral gable $18$ in on a side.

• Window's diagonal = ______ in
• Gable's height = ______ in
• $\tfrac{18}{\sqrt{2}}$ simplifies to ______ ; half the diagonal is ______ ; do they agree? ______ Why: ____________________
• To the nearest hundredth: ______ in and ______ in. Why the exact form was written first: ____________________

Review 3 (G.TR.4 b, c, f). Two right triangles, each with hypotenuse $20$ cm: one with a $30°$ angle, one with two $45°$ angles.

• $30°$-$60°$-$90°$: legs ______ and ______
• $45°$-$45°$-$90°$: legs ______ and ______ Rationalizing: ____________________
• Which has the longer shortest side? ______ Exact support: ____________________
• Pythagorean check for each: ____________________ What the check catches that the ratio did not: ____________________

Canva production notes

• Page size: 8.5 × 11 in, 0.6 in margins. One workbook page per Canva page.
• Type: page title H1 28 pt, section label H2 18 pt, body 12 pt, answer blanks 12 pt with a 1 pt rule.
• Figures: place at the width noted beside each FIGURE: line. All figures are 200 dpi PNG on white.
• Radical blanks need height. Most answers on pages 17 and 22 are radicals, and several are fractions with a radical numerator — $\tfrac{10\sqrt{3}}{3}$, $\tfrac{5\sqrt{2}}{2}$. Set those blanks at least 1.8 line-heights tall so a stacked fraction with an overbar fits without collision.
• Two blanks per item on pages 17 and 22. Items that ask for two sides need two clearly separated blanks with the side named above each; a single long rule invites one answer where two were wanted.
• Rationalizing must be shown, not just answered. Items 71, 93, and Review 2 and 3 each need a full ruled line for the working, not a value blank — the step is what the standard is asking for.
• The blank frames on page 24 are fig14-blank-right-triangle-frames.png. Reprint for any extra right-triangle problem a teacher assigns.
• Symbols: √, ², °, ≈, and the fraction bar must render in the body font; check the Canva font supports them before the first export. The degree signs in "45°-45°-90°" must not be dropped — the name of the triangle is its angles.
• Item numbers are continuous from 1 to 110 and must not be renumbered when pages are reordered.