Question

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

Explanation:

Sub - question 1: Area of the smallest square

Step 1: Identify side length

From the diagram, the side length of the smallest square is 6 (assuming the label 6 is for its side).

Step 2: Calculate area

The area of a square is given by \( A = s^2 \), where \( s \) is the side length. So for \( s = 6 \), \( A=6^2 = 36 \).

Step 1: Identify side length

From the diagram, the side length of the medium - sized square is also 6 (assuming the other 6 label is for its side).

Step 2: Calculate area

Using the formula for the area of a square \( A = s^2 \), with \( s = 6 \), we get \( A = 6^2=36 \)? Wait, no, maybe I misread. Wait, if the right triangle has legs of length 6 and 8? Wait, maybe the medium - sized square has side length 8? Wait, the diagram shows a right triangle with legs 6 and 8? Wait, maybe the smallest square has side 6, medium has side 8. Let's re - evaluate. If the right triangle has legs \( a = 6 \) and \( b = 8 \), then the area of the square on the leg of length 6 is \( 6^2 = 36 \), and on the leg of length 8 is \( 8^2=64 \). Maybe the smallest square has side 6, area \( 6\times6 = 36 \), medium has side 8, area \( 8\times8 = 64 \). Let's correct.

Step 1: Assume side length

If the medium - sized square has side length 8 (from the right triangle leg), then area \( A=s^2\), \( s = 8 \), so \( A = 8^2=64 \).

Step 1: Apply Pythagorean theorem

For a right triangle, the Pythagorean theorem states that \( c^2=a^2 + b^2 \), where \( c \) is the hypotenuse and \( a,b \) are the legs. The area of the square on the hypotenuse (the largest square) is equal to \( c^2 \), and the areas of the squares on the legs are \( a^2 \) and \( b^2 \). From sub - questions 1 and 2, if \( a^2 = 36 \) and \( b^2=64 \), then \( c^2=a^2 + b^2=36 + 64 = 100 \).

Answer:

36

Sub - question 2: Area of the medium - sized square