A triangle is drawn and squares are created from the lengths of each side.
Square A has an area of 14 units² and Square B has an area of 60 units².
What must be the area of the 3rd square for triangle to have a right angle? Click Here for Help Video.
(10 points)
The area for square C is
units².
To determine the area of square C, we need to find the length of the third side of the triangle.
Let's suppose the sides of the triangle are a, b, and c.
Square A has an area of 14, so its side length is √14.
Square B has an area of 60, so its side length is √60.
By the Pythagorean theorem, a^2 + b^2 = c^2, where c is the hypotenuse (the longest side) of the right triangle.
Substituting the side lengths of squares A and B, we have (√14)^2 + (√60)^2 = c^2.
This simplifies to 14 + 60 = c^2.
Therefore, c^2 = 74.
Now, we need to find the area of square C, which is the square of the side length of c.
So, the area of square C is c^2 = 74 units².
To determine the area of square C, we need to find the length of the third side of the triangle.
Squares are created from the lengths of each side of the triangle, so the sides of triangle are the square roots of the areas of the corresponding squares.
Given that square A has an area of 14 units², the length of its side is √14 units.
Similarly, square B has an area of 60 units², so the length of its side is √60 units.
For triangle with a right angle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the hypotenuse is the side with an unknown length, which we will call x.
According to the Pythagorean theorem, we have:
x² = (√14)² + (√60)²
Simplifying this equation gives us:
x² = 14 + 60
x² = 74
Taking the square root of both sides, we find:
x = √74 units
Now that we know the length of the third side, we can determine the area of square C by squaring this length.
The area of square C is:
(√74)² = 74 units²
Therefore, the area for square C is 74 units².
To find the area of square C, we need to determine the length of its side.
First, let's find the lengths of the sides of the triangle.
The area of square A is given as 14 units². Since the area of a square is the square of its side length, we can calculate the side length of square A by taking the square root of 14: √14 ≈ 3.74 units.
Similarly, the area of square B is given as 60 units². Taking the square root of 60, we find that the side length of square B is √60 ≈ 7.75 units.
Now, we have the lengths of two sides of the triangle. Let's label them as side a (3.74 units) and side b (7.75 units).
To determine if the triangle has a right angle, we can use the Pythagorean theorem, which 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 other two sides.
Let's calculate the sum of squares of side a and side b:
a² + b² = 3.74² + 7.75²
≈ 13.96 + 60.06
≈ 74.02
To find the length of the hypotenuse, we take the square root of the sum of squares:
c ≈ √74.02 ≈ 8.6 units.
Now, we can find the area of square C by squaring the length of the hypotenuse:
Area of square C = c² = 8.6² ≈ 73.96 units².
Therefore, the area of square C should be approximately 73.96 units² for the triangle to have a right angle.