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 third square for the triangle to have a right angle?
Let the lengths of the sides of the triangle be a, b, and c.
Since Square A has an area of 18 units², its side length is √18 = 3√2 units.
Since Square B has an area of 90 units², its side length is √90 = 3√10 units.
By Pythagoras' theorem, in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides.
So, c² = a² + b²
(3√10)² = (3√2)² + a²
90 = 18 + a²
a² = 90 - 18 = 72
a = √72 = 6√2 units
Now, the third square must have the side length of a in order for the triangle to have a right angle.
Therefore, the area of the third square is a² = (6√2)² = 72 units². Answer: \boxed{72}.
To find the area of the third square, we need to determine the length of the corresponding side of the triangle.
Let's assume the lengths of the sides of the triangle are a, b, and c, where a and b are the legs of the triangle and c is the hypotenuse.
We know that the area of square A is given by the formula: Area_A = side_a^2 = 18 units².
Therefore, side_a = sqrt(Area_A) = sqrt(18) units.
Similarly, the area of square B is given by the formula: Area_B = side_b^2 = 90 units².
Thus, side_b = sqrt(Area_B) = sqrt(90) units.
For a right-angled triangle, the Pythagorean theorem states that a^2 + b^2 = c^2.
We can substitute the values we found earlier to get the equation sqrt(18)^2 + sqrt(90)^2 = c^2.
Simplifying this equation, we have 18 + 90 = c^2.
Therefore, c^2 = 108.
To find the length of side_c, we calculate the square root of 108: side_c = sqrt(108) units.
Now, for the triangle to have a right angle, the side lengths a, b, and c must satisfy the Pythagorean theorem; and for the third square, we need the area of its side to be equal to side_c^2.
Thus, the area of the third square should be side_c^2 = (sqrt(108))^2 = 108 units².
To determine the area of the third square, we need to find the length of the corresponding side of the triangle.
Let's denote the sides of the triangle as a, b, and c, with a being the side corresponding to square A, b being the side corresponding to square B, and c being the unknown side corresponding to the third square.
We know that the area of a square is given by side length squared, thus we can find the side lengths of square A and square B by taking the square root of their respective areas:
Side length of square A = √(Area of square A) = √18 = 3√2
Side length of square B = √(Area of square B) = √90 = 3√10
Now, let's examine the relationship between the sides of a right-angled triangle. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (the longest side, denoted by c) is equal to the sum of the squares of the other two sides (a and b):
c² = a² + b²
Substituting the values we obtained:
c² = (3√2)² + (3√10)²
c² = 18 + 90
c² = 108
To find the side length of the third square (c), we need to take the square root of 108:
Side length of square C = √108 = 6√3
Finally, we can calculate the area of the third square by squaring its side length:
Area of square C = (Side length of square C)² = (6√3)² = 36 * 3 = 108 units²
Therefore, the area of the third square must be 108 square units for the triangle to have a right angle.
You are cleaning the gutters out of your house which stands 12 feet tall. You are using a ladder that is 20 feet long to reach the gutters. How far away from the house is the base if the ladder?
To determine how far away from the house the base of the ladder is, we can use the Pythagorean theorem.
The height of the house is the height of the right triangle, which is 12 feet.
The length of the ladder is the hypotenuse of the right triangle, which is 20 feet.
Let's denote the distance from the base of the ladder to the house as x.
According to the Pythagorean theorem, the sum of the squares of the two shorter sides (a and b) is equal to the square of the hypotenuse (c).
a² + b² = c²
We can plug in the values we have:
12² + x² = 20²
144 + x² = 400
To solve for x, we subtract 144 from both sides:
x² = 400 - 144
x² = 256
Now, we take the square root of both sides:
x = √256
x = 16
Therefore, the base of the ladder is 16 feet away from the house.
A rectangular park that measures 40 yards by 80 yards will have a diagonal path built. How long will the diagonal path be? Round to the nearest tenth.
To find the length of the diagonal path in the rectangular park, 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 diagonal) is equal to the sum of the squares of the lengths of the other two sides (the width and length of the rectangle).
In this case, the width of the rectangle is 40 yards and the length is 80 yards. Let's denote the diagonal as d.
According to the Pythagorean theorem, we have:
d² = 40² + 80²
d² = 1600 + 6400
d² = 8000
To find the length of the diagonal, we take the square root of both sides:
d = √8000
d ≈ 89.4 yards
Therefore, the length of the diagonal path in the rectangular park will be approximately 89.4 yards, rounded to the nearest tenth.