Pythagorean Theorem Worksheet featuring 8 right triangle problems to solve for unknown sides using the Pythagorean theorem.

I'll solve each problem using the Pythagorean Theorem: a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse (the longest side opposite the right angle).

Let me work through each problem carefully:

Problem 1: Given legs 8 cm and 6 cm, find hypotenuse c
- a = 8, b = 6
- c² = 8² + 6² = 64 + 36 = 100
- c = √100 = 10 cm

Problem 2: Given leg 24 yd and hypotenuse 26 yd, find leg a
- b = 24, c = 26
- a² + 24² = 26²
- a² + 576 = 676
- a² = 676 - 576 = 100
- a = √100 = 10 yd

Problem 3: Given leg 5 cm and hypotenuse 13 cm, find leg b
- a = 5, c = 13
- 5² + b² = 13²
- 25 + b² = 169
- b² = 169 - 25 = 144
- b = √144 = 12 cm

Problem 4: Given leg 11 in and hypotenuse 12 in, find leg a
- b = 11, c = 12
- a² + 11² = 12²
- a² + 121 = 144
- a² = 144 - 121 = 23
- a = 23 in (exact answer)

Problem 5: Given legs 3 m and 4 m, find hypotenuse x
- a = 3, b = 4
- x² = 3² + 4² = 9 + 16 = 25
- x = √25 = 5 m

Problem 6: Given leg 6 mm and hypotenuse 10 mm, find leg x
- b = 6, c = 10
- x² + 6² = 10²
- x² + 36 = 100
- x² = 100 - 36 = 64
- x = √64 = 8 mm

Problem 7: Rectangle with diagonal 15 ft and one side 12 ft, find other side x
The diagonal forms a right triangle with the sides of the rectangle.
- One leg = 12, hypotenuse = 15
- x² + 12² = 15²
- x² + 144 = 225
- x² = 225 - 144 = 81
- x = √81 = 9 ft

Problem 8: Trapezoid - need to find the slanted side x
Looking at this shape, I can see it's a trapezoid with:
- Left vertical side = 6
- Top horizontal side = 3
- Bottom horizontal side = 10

To find x, I need to create a right triangle. The horizontal difference is 10 - 3 = 7, and the vertical height is 6.
- So we have a right triangle with legs 6 and 7
- x² = 6² + 7² = 36 + 49 = 85
- x = √85 (exact answer)

Final Answer:
1. c = 10 cm
2. a = 10 yd
3. b = 12 cm
4. a = √23 in
5. x = 5 m
6. x = 8 mm
7. x = 9 ft
8. x = √85