Use the image to answer the question.
An illustration shows a square pyramid with labeled parts. The left, front and right faces of the pyramid and base are visible. The back face is not visible. The back base edge is not visible but its position is indicated by a dashed line. The front base edge shows a length of 40 feet. A dotted line extends from the interior of the vertex straight down to the center of the base and is labeled h. A right angle is indicated where the dotted height line meets the base. Another dotted line extends from the center of the base where the height line meets to the center of the right base edge. A dotted line extends from the right base edge straight up the center of the right face of the prism and meets the height line at the vertex. This dotted line shows a slant height of 25 feet.
Apply the Pythagorean Theorem to determine the height of the square pyramid, h . Round the answer to the nearest tenth.
To apply the Pythagorean Theorem, we need to find the lengths of the other two sides of the right triangle formed by the height line (h), the slant height, and the base of the square pyramid.
The slant height is given as 25 feet. The length of the base (one side of the square) is given as 40 feet.
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.
Hence, using the Pythagorean Theorem, we can find the height (h).
Let's call the other two sides of the right triangle x and y.
According to the theorem:
x^2 + y^2 = 25^2
We also know that the length of the base (one side of the square) is 40 feet, and since the pyramid is a right square pyramid, the base is a square.
So, x = y = 40 feet.
Substituting the values into the equation:
2x^2 = 25^2
2(40)^2 = 625
2(1600) = 625
3200 = 625
3200/2 = 625/2
1600 = 312.5
Taking the square root of both sides:
√1600 = √312.5
40 = √312.5
Therefore, the height of the square pyramid, h, is approximately 40 feet.
We can use the Pythagorean Theorem to determine the height of the square pyramid, h. The Pythagorean Theorem 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 the given illustration, the height line, the line from the center of the base to the vertex, and the slant height form a right triangle. The slant height is 25 feet, and the base length is 40 feet.
Let's label the sides of the right triangle:
- The slant height: a = 25 feet
- The base length: b = 40 feet
- The height: h (what we want to find)
Using the Pythagorean Theorem equation, we have:
a^2 = b^2 + h^2
Substituting the given values:
25^2 = 40^2 + h^2
625 = 1600 + h^2
Rearranging the equation:
h^2 = 625 - 1600
h^2 = -975
Since we cannot take the square root of a negative number, it means that there is no real solution for the height h.
To determine the height of the square pyramid, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right-angled 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 this case, the slant height of the pyramid (25 feet) is the hypotenuse, and the height of the pyramid (h) and half of the length of the base (20 feet) are the other two sides. So we can set up the equation as follows:
h^2 + 20^2 = 25^2
Simplifying this equation, we get:
h^2 + 400 = 625
Subtracting 400 from both sides, we have:
h^2 = 225
Taking the square root of both sides, we get:
h = √225
h = 15 feet
Therefore, the height of the square pyramid is 15 feet.