Question

find the areas and side lengths of the squares

1. the big square is formed by a small square and 4 congruent isosceles right triangles.

a = 289 square units
12 units
small square big square
area
(square units)
side length
(units)

Explanation:

Step1: Find side - length of big square

Given the area of the big square $A_{big}=289$ square units. Using the formula for the area of a square $A = s^{2}$, where $s$ is the side - length. Solving for $s$ gives $s_{big}=\sqrt{A_{big}}=\sqrt{289}=17$ units.

Step2: Analyze the right - triangle

In the isosceles right - triangle, the non - hypotenuse side length of each isosceles right - triangle is $12$ units.

Step3: Find side - length of small square

The side - length of the small square $s_{small}$ can be found by considering the relationship between the big square and the triangles. The side - length of the big square is equal to the sum of the lengths of the non - hypotenuse sides of two adjacent isosceles right - triangles. So $s_{small}=s_{big}-2\times12$. Substituting $s_{big} = 17$ units, we get $s_{small}=17 - 2\times12=17 - 24 = 5$ units.

Step4: Find area of small square

Using the area formula $A = s^{2}$, for the small square with $s = 5$ units, $A_{small}=s_{small}^{2}=5^{2}=25$ square units.

Answer:

	Small Square	Big Square
Side Length (units)	5	17