In a visual proof, after rearranging four identical right triangles (legs $a, b$) inside a large square, the remaining uncovered space forms two smaller squares. What are the areas of these two squares?

$(\frac{a}{2})^2$ and $(\frac{b}{2})^2$

Understand the Pythagorean Theorem Practice — Grade 8 math

Lesson 1: Understand the Pythagorean Theorem — Practice Questions

1. Four identical 5-12-13 right triangles are arranged in a large square with side length 17. This leaves a smaller, tilted square in the center. What is the area of this central square? ___
2. In a visual proof, after rearranging four identical right triangles (legs $a, b$) inside a large square, the remaining uncovered space forms two smaller squares. What are the areas of these two squares?
- A. $a^2$ and $b^2$
- B. $c^2$ and $a^2$
- C. $(a+b)^2$ and $c^2$
- D. $(\frac{a}{2})^2$ and $(\frac{b}{2})^2$
3. A visual proof of the Pythagorean theorem uses four identical right triangles with legs of 8 cm and 15 cm. What is the total area of the four triangles combined? ___
4. For a visual proof of $a^2 + b^2 = c^2$, four identical right triangles with legs $a$ and $b$ are placed inside a larger square. What is the side length of this large outer square?
- A. $a+b$
- B. $c$
- C. $\sqrt{a^2+b^2}$
- D. $2a+2b$
5. A large square with side length 31 inches contains four identical right triangles with legs 7 and 24 inches. What is the side length of the inner tilted square formed by the hypotenuses? ___
6. A rectangular park is 24 meters long and 7 meters wide. If you walk diagonally from one corner to the opposite one, what is the distance you walk, in meters? The distance is ___ meters.
7. A 41-foot ladder leans against a building. The base of the ladder is 9 feet from the wall. How high up the building does the ladder reach, in feet? The height is ___ feet.
8. The height of a computer monitor is 6 inches and its width is 10 inches. What is the length of the diagonal of the monitor, rounded to the nearest tenth of an inch?
- A. 11.7 inches
- B. 8.0 inches
- C. 16.0 inches
- D. 136.0 inches
9. A kite is flying on a 15-meter string. The kite is directly above a point 8 meters away from the person holding the string. What is the height of the kite, rounded to the nearest meter?
- A. 7 meters
- B. 13 meters
- C. 17 meters
- D. 161 meters
10. A triangular sail has a base of 11 feet and a height of 60 feet, forming a right angle. What is the length of the sail's longest edge, in feet? The length is ___ feet.