Question

task 3
here, we have a right triangle with squares built off its sides.
given the information in the diagram above:

1. what is the area of the smallest square?
2. what is the area of the largest square?
3. given what you see here in this geogebra app, what would the area of the medium - sized square be?
4. given your answer to (3), what would the side length (x) of the medium - sized square be?

Explanation:

1)

Step1: Identify side of smallest square

The smallest square is built on the side of length 5. For a square, area = side².
$Area = 5^2 = 25$

Step1: Identify side of largest square

The largest square is built on the side of length 13. Area of square = side².
$Area = 13^2 = 169$

Step1: Recall Pythagorean theorem for areas

In a right triangle, the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse. Let \( A_{small} = 25 \), \( A_{large} = 169 \), and \( A_{medium} \) be the area of the medium - sized square. Then \( A_{small}+A_{medium}=A_{large} \) (wait, no, actually the two legs' squares sum to the hypotenuse's square. Wait, the right triangle has legs 5 and \( x \), hypotenuse 13? Wait, no, looking at the diagram, the right triangle has one leg 5, hypotenuse 13, and the other leg \( x \). Wait, no, the squares: the smallest square (side 5), medium square (side \( x \)), largest square (side 13). By Pythagorean theorem, \( 5^{2}+x^{2}=13^{2} \), so the area of the medium square \( A = x^{2}=13^{2}-5^{2} \).

Step2: Calculate \( 13^{2}-5^{2} \)

\( 13^{2}=169 \), \( 5^{2}=25 \), so \( 169 - 25=144 \)

Answer:

25

2)