A triangle is drawn and squares are created from the lengths of each side.
Square A has an area of 25 units and Square C has an area of 70 units
What must be the area of the 3rd square for triangle to have a right angle?
To determine the area of the third square for the triangle to have a right angle, we need to find the length of the third side of the triangle first.
Let's assume the side lengths of the triangle are a, b, and c.
The area of square A is given as 25 units, which means the side length of square A is √25 = 5 units. Thus, side a of the triangle is 5 units.
The area of square C is given as 70 units, which means the side length of square C is √70 units. Since squares A and C are created from the lengths of each side of the triangle, side a is the same as the side length of square A (5 units). Therefore, side c of the triangle is √70 units.
We want to find the side length of the third side, side b, in order to calculate the area of the third square.
In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
According to the Pythagorean theorem: a^2 + b^2 = c^2
Substituting the known values: 5^2 + b^2 = (√70)^2
25 + b^2 = 70
b^2 = 70 - 25
b^2 = 45
b = √45 = √9 * √5 = 3√5
Therefore, the length of side b is 3√5 units.
Now, to calculate the area of the third square:
The area of square B is equal to the square of the side length, so the area of square B is (side b)^2.
Area of square B = (3√5)^2 = 9 * 5 = 45 units.
Therefore, the area of the third square (square B) must be 45 units for the triangle to have a right angle.