Question

ih. three squares are joined below at their vertices to form a right triangle.
find the value of x, the side length of the square marked in the diagram.
225 ft²
81ft²
x

Explanation:

Step1: Identify the theorem

This is a right triangle with squares on its sides, so we use the Pythagorean theorem. Let the areas of the squares be \( A_1 = x^2 \), \( A_2 = 81 \, \text{ft}^2 \), and \( A_3 = 225 \, \text{ft}^2 \). By the Pythagorean theorem, \( A_1 + A_2 = A_3 \) (since the triangle is right - angled, the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse). Wait, actually, if the right triangle has legs with lengths corresponding to the sides of the squares with areas \( x^2 \) and \( 81 \), and hypotenuse corresponding to the square with area \( 225 \), then \( x^2+81 = 225 \).

Step2: Solve for \( x^2 \)

Subtract 81 from both sides of the equation \( x^2 + 81=225 \). So \( x^2=225 - 81 \). Calculate \( 225-81 = 144 \).

Step3: Solve for \( x \)

Take the square root of both sides. Since \( x \) represents a length, we take the positive square root. \( x=\sqrt{144}=12 \).

Answer:

The value of \( x \) is \( 12 \) feet.