A triangle is drawn and squares are created from the lengths of each side. Square A has an area of 18 units 2 and Square B has an area of 90 units2 . What must be the area of the 3rd square for triangle to have a right angle?
To find the area of the third square, we need to find the length of the third side of the triangle. Let's assume the side lengths of the triangle are a, b, and c, and the areas of the squares are A, B, and C respectively.
We know that the area of a square is equal to the square of its side length. So we have:
A = a^2
B = b^2
C = c^2
We are given the areas of squares A and B:
A = 18 units^2
B = 90 units^2
We can set up an equation using the Pythagorean theorem:
a^2 + b^2 = c^2
Substituting the areas of the squares, we get:
18 + 90 = c^2
108 = c^2
Taking the square root of both sides, we find:
c = √108
Simplifying, we get:
c = 6√3 units
To find the area of the third square, we square the side length:
C = (6√3)^2
C = 36 * 3
C = 108 units^2
Therefore, the area of the third square must be 108 square units for the triangle to have a right angle.
To determine the area of the third square in order for the triangle to have a right angle, we need to determine the side lengths of the triangle.
Let's assume that the sides of the triangle are a, b, and c, and the areas of the squares are A, B, and C, respectively.
Square A has an area of 18 units^2, so the side length of square A is √18 = 3√2 units.
Square B has an area of 90 units^2, so the side length of square B is √90 = 3√10 units.
Knowing the side lengths of squares A and B, we can determine the side lengths of the triangle. Since the sides of the triangle are the same lengths as the sides of the squares, we can conclude that:
Side length a = 3√2 units
Side length b = 3√10 units
In a right-angled triangle, the squares of the lengths of the two shorter sides add up to the square of the length of the longest side. Therefore, we can calculate the area of the third square (C) as:
C = a^2 + b^2
C = (3√2)^2 + (3√10)^2
C = 18 + 90
C = 108 units^2
Therefore, the area of the third square must be 108 units^2 for the triangle to have a right angle.
To determine the area of the third square, we need to find the side length of the triangle. Since the squares are created from the lengths of each side, we can find the side lengths of the triangle by taking the square roots of the areas of the squares.
For Square A with an area of 18 units^2, we can find the side length by taking the square root of 18:
Side length of Square A = √18
Similarly, for Square B with an area of 90 units^2, we can find the side length by taking the square root of 90:
Side length of Square B = √90
To form a triangle with a right angle, we can use the Pythagorean theorem, which states that 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. In our case, the side lengths of Square A and Square B can be the two sides, and the side length of the third square can be the hypotenuse.
Therefore, we have the equation:
(Side length of Square A)^2 + (Side length of Square B)^2 = (Side length of the third square)^2
(√18)^2 + (√90)^2 = (Side length of the third square)^2
Simplifying:
18 + 90 = (Side length of the third square)^2
108 = (Side length of the third square)^2
To find the side length of the third square, we take the square root of both sides:
√108 = Side length of the third square
By simplifying, we get:
Side length of the third square = √(2^2 * 3^3) = 6√3
Finally, to find the area of the third square, we square the side length:
Area of the third square = (Side length of the third square)^2 = (6√3)^2 = 36(√3)^2 = 36 * 3 = 108 units^2
Therefore, the area of the third square must be 108 units^2 for the triangle to have a right angle.