To find the area of the third square, which is on the hypotenuse of the right triangle, we can use the Pythagorean theorem. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's denote the side lengths of the smaller squares as "a" for the square with area 5 cm², and "b" for the square with area 12 cm². The side length of the third square (on the hypotenuse) is denoted as "c."
We are given:
- The area of square 'a' which is [tex]\( area_a = a^2 = 5 \text{ cm}^2 \)[/tex]
- The area of square 'b' which is [tex]\( area_b = b^2 = 12 \text{ cm}^2 \)[/tex]
The side lengths of the squares are the square roots of their areas, therefore:
- Side length 'a' is [tex]\( a = \sqrt{area_a} = \sqrt{5} \text{ cm} \)[/tex]
- Side length 'b' is [tex]\( b = \sqrt{area_b} = \sqrt{12} \text{ cm} \)[/tex]
According to the Pythagorean theorem, the sum of the squares of side lengths 'a' and 'b' will be equal to the square of side length 'c':
[tex]\[ c^2 = a^2 + b^2 \][/tex]
Substitute the values of [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] from the information given:
[tex]\[ c^2 = 5 \text{ cm}^2 + 12 \text{ cm}^2 \][/tex]
[tex]\[ c^2 = 17 \text{ cm}^2 \][/tex]
Therefore, the area of the third square (which is on the hypotenuse) is [tex]\( c^2 \)[/tex], which is 17 cm².
