Pythagorean Theorem — Geometry 练习题

Grades 8–10 中等难度 10 道题

The Pythagorean theorem states that in a right triangle with legs aa and bb and hypotenuse cc, a2+b2=c2a^2 + b^2 = c^2. It applies only to right triangles — triangles containing a 90°90° angle.

The hypotenuse is always the longest side, opposite the right angle. When finding a leg, rearrange to a2=c2−b2a^2 = c^2 - b^2. Always take the positive square root for side lengths.

The converse helps determine if a triangle is right: if a2+b2=c2a^2 + b^2 = c^2, the triangle is a right triangle. This is useful in construction and navigation.

练习的技能

  • Identifying hypotenuse and legs
  • Finding missing side lengths
  • Applying the theorem to word problems
  • Using the converse to test for right triangles
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练习题：Pythagorean Theorem

说明： 仔细解答每一道题。清晰地展示所有解题过程。在指定位置或按指示在另一张纸上写出最终答案。

  1. 1.A right triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.

Pythagorean Theorem — 练习题（续）

  1. 2.A 13-foot ladder is leaning against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
  2. 3.Find the missing side length of a right triangle with a hypotenuse of 25 inches and one leg of 7 inches.

Pythagorean Theorem — 练习题（续）

  1. 4.A rectangular field is 40 meters long and 30 meters wide. What is the distance from one corner to the opposite corner?
  2. 5.The legs of a right triangle are 9 cm and 12 cm. What is the perimeter of the triangle?

Pythagorean Theorem — 练习题（续）

  1. 6.A television screen measures 32 inches diagonally and 24 inches in height. What is the width of the screen?
  2. 7.A right triangle has a hypotenuse of 17 feet and one leg of 15 feet. Find the length of the other leg.

Pythagorean Theorem — 练习题（续）

  1. 8.A ramp rises 3 feet over a horizontal distance of 4 feet. What is the length of the ramp?
  2. 9.Two friends start at the same point. One walks 5 miles north, the other walks 12 miles east. How far apart are they?

Pythagorean Theorem — 练习题（续）

  1. 10.A right triangle has legs of lengths xx and x+1x+1, and a hypotenuse of length 5. Find the value of xx.

参考答案

  1. 1.
    c=62+82=36+64=100=10c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 cm
    最终答案： 10 cm
  2. 2.
    h=132−52=169−25=144=12h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 feet
    最终答案： 12 feet
  3. 3.
    b=252−72=625−49=576=24b = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 inches
    最终答案： 24 inches
  4. 4.
    d=402+302=1600+900=2500=50d = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 meters
    最终答案： 50 meters
  5. 5.
    Hypotenuse =92+122=81+144=225=15= \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 cm. Perimeter =9+12+15=36= 9 + 12 + 15 = 36 cm
    最终答案： 36 cm
  6. 6.
    w=322−242=1024−576=448≈21.2w = \sqrt{32^2 - 24^2} = \sqrt{1024 - 576} = \sqrt{448} \approx 21.2 inches
    最终答案： 21.2 inches

参考答案（续）

  1. 7.
    b=172−152=289−225=64=8b = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8 feet
    最终答案： 8 feet
  2. 8.
    r=32+42=9+16=25=5r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 feet
    最终答案： 5 feet
  3. 9.
    d=52+122=25+144=169=13d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 miles
    最终答案： 13 miles
  4. 10.
    x2+(x+1)2=52x^2 + (x+1)^2 = 5^2; x2+x2+2x+1=25x^2 + x^2 + 2x + 1 = 25; 2x2+2x−24=02x^2 + 2x - 24 = 0; x2+x−12=0x^2 + x - 12 = 0; (x+4)(x−3)=0(x+4)(x-3)=0; x=3x=3 (positive solution)
    最终答案： 3
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需要避免的常见错误

  • Adding legs when the hypotenuse is needed (should subtract)
  • Forgetting the square root at the end
  • Applying the theorem to non-right triangles
  • Using the wrong side as the hypotenuse

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