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? A standard proof image of the Pythagorean theorem. (5 points)
The area of Square C=
units2
Triangle ABC is a right triangle, where AB and BC are the two legs and AC is the hypotenuse.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Let's say the length of side AB is x. Then, the length of side BC is also x, since they are both legs of the triangle.
Square A has an area of 18 units^2. Since the length of one side is x, we can write the equation:
x^2 = 18
Square B has an area of 90 units^2. Since the length of one side is x, we can write the equation:
x^2 = 90
Now, we have a system of two equations:
x^2 = 18 ...(1)
x^2 = 90 ...(2)
From equation (1), we can solve for x:
x^2 = 18
x = sqrt(18)
x ≈ 4.2426
From equation (2), we can solve for x:
x^2 = 90
x = sqrt(90)
x ≈ 9.4868
Now, to find the area of square C, which is the square of the hypotenuse AC, we can use the Pythagorean theorem:
AC^2 = AB^2 + BC^2
Substituting the values we found:
AC^2 = (4.2426)^2 + (9.4868)^2
AC^2 ≈ 18 + 90
AC^2 ≈ 108
Therefore, the area of Square C is 108 units^2.
To find the area of Square C and determine if the triangle has a right angle, we can use the Pythagorean theorem.
The Pythagorean 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. Mathematically, it can be represented as:
c^2 = a^2 + b^2
Where:
c = length of the hypotenuse
a = length of one side
b = length of the other side
In this case, let's assume that the sides of the triangle correspond to the sides of Square A, Square B, and Square C.
Square A has an area of 18 units^2, so its side length (denoted as a) can be found by taking the square root of 18:
a = √18
Similarly, Square B has an area of 90 units^2, so its side length (denoted as b) can be found by taking the square root of 90:
b = √90
To determine the area of Square C, we need to find its side length (denoted as c) using the Pythagorean theorem:
c^2 = a^2 + b^2
Substituting the values we found earlier:
c^2 = (√18)^2 + (√90)^2
Simplifying:
c^2 = 18 + 90
c^2 = 108
To find the side length of Square C (c), we need to take the square root of 108:
c = √108
Finally, to find the area of Square C, we square the side length:
Area of Square C = c^2
Area of Square C = (√108)^2
Area of Square C = 108 units^2
Therefore, the area of Square C must be 108 units^2 for the triangle to have a right angle.
To solve this problem, we'll 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 assume that Square A corresponds to the shorter leg of the triangle, while Square B corresponds to the longer leg, and Square C corresponds to the hypotenuse.
We know that the area of Square A is 18 units², and the area of Square B is 90 units².
The formula to calculate the area of a square is A = s², where A is the area and s is the length of the side.
By substituting the given areas of the squares into the formula, we can find the lengths of the sides of each square.
For Square A:
18 = s^2
Taking the square root of both sides, we get:
s = √18 = 3√2 units
For Square B:
90 = s^2
Taking the square root of both sides, we get:
s = √90 = 3√10 units
Now that we know the lengths of the sides of Square A and Square B, we can find the length of the hypotenuse, which is the side of Square C.
According to the Pythagorean theorem, the sum of the squares of the lengths of the legs (Square A and Square B) is equal to the square of the length of the hypotenuse (Square C):
(Square A length)^2 + (Square B length)^2 = (Square C length)^2
(3√2 units)^2 + (3√10 units)^2 = (Square C length)^2
18 + 90 = (Square C length)^2
108 = (Square C length)^2
To find the area of Square C, we just need to square the length of the side:
Area of Square C = (Square C length)^2 = 108 units²
Therefore, the area of Square C must be 108 units² for the triangle to have a right angle.