a triangle is drawn and squares are created from the lengths of each side. Square A has an area of 18 units2 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? (5 points)
To determine the area of the third square, we need to find the length of its side.
Let the three sides of the triangle be a, b, and c, and let the lengths of the sides of the squares A, B, and C be x, y, and z respectively.
According to the given information, we have the following equations:
Area of square A = x^2 = 18
Area of square B = y^2 = 90
Since the areas of squares A and B are given, we can solve these equations to find the lengths of their sides:
x^2 = 18
x = √18
x ≈ 4.24
y^2 = 90
y = √90
y ≈ 9.49
Now let's consider square C. To have a right angle in the triangle, the sum of the areas of squares A and B should be equal to or greater than the area of square C.
x^2 + y^2 ≥ z^2
Substituting the values of x and y, we get:
4.24^2 + 9.49^2 ≥ z^2
18 + 90 ≥ z^2
108 ≥ z^2
To determine the minimum area of square C, we need to take the square root of both sides:
√(108) ≥ z
z ≥ √108
z ≥ 10.39
Therefore, the minimum area of the third square, area C, would be approximately 10.39 units^2 for the triangle to have a right angle.
To determine the area of the third square, we need to find the length of the third side of the triangle. Let's call the length of the first side of the triangle "a," the length of the second side "b," and the length of the third side "c."
Since Square A has an area of 18 units^2, the side length of Square A will be the square root of 18:
Side length of Square A = √18
Similarly, since Square B has an area of 90 units^2, the side length of Square B will be the square root of 90:
Side length of Square B = √90
Now, let's determine the lengths of sides a, b, and c. According to the Pythagorean theorem, in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
a^2 + b^2 = c^2
The square of the side length of Square A (a) plus the square of the side length of Square B (b) must be equal to the square of the side length of the third square (c):
(√18)^2 + (√90)^2 = c^2
18 + 90 = c^2
108 = c^2
Now, taking the square root of both sides, we get:
√108 = c
Simplifying, we find:
6√3 = c
So, the side length of the third square is 6√3 units. To find the area of the square, we square this length:
Area of the third square = (6√3)^2
Area of the third square = 36 * 3
Area of the third square = 108 units^2
Therefore, the area of the third square must be 108 units^2 for the triangle to have a right angle.
To solve this problem, we need to understand the relationship between the area of a square and the lengths of its sides, as well as the relationship between the sides of a right-angled triangle.
First, let's consider the relationship between the area of a square and the lengths of its sides. In a square, all sides are equal in length. Therefore, we can determine the length of the sides of a square by taking the square root of its area.
For Square A with an area of 18 units², we can find the length of its sides by taking the square root of 18:
√18 ≈ 4.24 (rounded to two decimal places)
For Square B with an area of 90 units², the length of its sides can be found by taking the square root of 90:
√90 ≈ 9.49 (rounded to two decimal places)
Next, let's consider the relationship between the sides of a right-angled triangle. According to the Pythagorean theorem, 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.
In our case, the sides of the triangle are the lengths of the sides of Square A and Square B. Let's label the sides of the triangle as follows:
Side A (opposite Square A) = 4.24
Side B (opposite Square B) = 9.49
Side C (hypotenuse) = ?
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
Side C² = Side A² + Side B²
Side C² = 4.24² + 9.49²
Side C² ≈ 18 + 90
Side C² ≈ 108
Now, to find the length of Side C, we take the square root of 108:
√108 ≈ 10.39 (rounded to two decimal places)
Since the triangle must have a right angle, we can create Square C with a side length of 10.39 units. The area of Square C can be calculated by squaring the length of its side:
Area of Square C = Side C² ≈ 10.39² ≈ 108 units²
Therefore, the area of the third square (Square C) should be approximately 108 units² in order for the triangle to have a right angle.