Math Problem Statement

Find the values of x in various right triangles using the Pythagorean Theorem and Pythagorean Triples.

Solution

The worksheet contains problems involving the Pythagorean Theorem and Pythagorean Triples. Let's solve each question in sequence.

Example 1: Using the Pythagorean Theorem

The Pythagorean Theorem states:
a2+b2=c2a^2 + b^2 = c^2 where cc is the hypotenuse of a right triangle, and aa and bb are the other two sides.

  1. Problem 1: Both legs are 3.
    32+32=x23^2 + 3^2 = x^2 9+9=x2x=18=324.249 + 9 = x^2 \Rightarrow x = \sqrt{18} = 3\sqrt{2} \approx 4.24

  2. Problem 2: Legs are 9 and 12; hypotenuse is xx. 92+122=x29^2 + 12^2 = x^2 81+144=x2x=225=1581 + 144 = x^2 \Rightarrow x = \sqrt{225} = 15

  3. Problem 3: One leg is 25, hypotenuse is 65, find the other leg xx. x2+252=652x^2 + 25^2 = 65^2 x2+625=4225x2=3600x=3600=60x^2 + 625 = 4225 \Rightarrow x^2 = 3600 \Rightarrow x = \sqrt{3600} = 60

  4. Problem 4: Legs are 5 and 9; hypotenuse is xx. 52+92=x25^2 + 9^2 = x^2 25+81=x2x=10610.3025 + 81 = x^2 \Rightarrow x = \sqrt{106} \approx 10.30

  5. Problem 5: Legs are 16 and 33; hypotenuse is xx. 162+332=x216^2 + 33^2 = x^2 256+1089=x2x=134536.68256 + 1089 = x^2 \Rightarrow x = \sqrt{1345} \approx 36.68

  6. Problem 6: Legs are 11 and 28; hypotenuse is xx. 112+282=x211^2 + 28^2 = x^2 121+784=x2x=90530.08121 + 784 = x^2 \Rightarrow x = \sqrt{905} \approx 30.08

Example 2: Using Pythagorean Triples

In this section, the problems involve recognizing patterns of Pythagorean triples, which are sets of three integers that satisfy the Pythagorean Theorem.

  1. Problem 7: Known triple 8,15,178, 15, 17; so x=15x = 15.

  2. Problem 8: Known triple 9,12,159, 12, 15, scaled by 5 to 45,60,x45, 60, x, so x=60x = 60.

  3. Problem 9: Known triple 7,24,257, 24, 25, scaled by 4 to 28,96,x28, 96, x, so x=100x = 100.

  4. Problem 10: Known triple 5,12,135, 12, 13; so x=13x = 13.

  5. Problem 11: Known triple 6,8,106, 8, 10; so x=6x = 6.

  6. Problem 12: Known triple 3,4,53, 4, 5, scaled by 4 to 12,16,x12, 16, x, so x=25x = 25.

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Math Problem Analysis

Mathematical Concepts

Geometry
Right Triangles
Pythagorean Theorem
Pythagorean Triples

Formulas

a^2 + b^2 = c^2

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10