Look at the square pyramid in the image shown. With the given measurements, determine the length of the base of the square pyramid. A square pyramid with a height of 20 miles, a slant height (diagonal) of 25 miles, and an unknown leg of a right triangle. The question is asking for the base of the square pyramid, not just x!
a2+b2=c2
(10 points)
The base length of the square pyramid is
miles long
To determine the length of the base of the square pyramid, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle with one side being the leg of the triangle (length of the base of the square pyramid), another side being the slant height (diagonal) of the pyramid, and the hypotenuse being the height of the pyramid.
Let's label the leg of the triangle as "x". Using the Pythagorean theorem, we have:
x^2 + 25^2 = 20^2
Simplifying this equation, we get:
x^2 + 625 = 400
Subtracting 625 from both sides, we have:
x^2 = 400 - 625
x^2 = -225
Taking the square root of both sides, we get:
x = ±√(-225)
Since we're looking for the length of the base, which cannot be negative, we disregard the negative solution. Therefore, x cannot be a real number solution in this case.
So, in this scenario, there is no real number solution for the length of the base of the square pyramid.
To solve this problem, we can use the Pythagorean theorem. Let's label the unknown leg of the right triangle as "x". We know that one leg is x, the other leg is the base of the square pyramid, and the hypotenuse is the slant height of the pyramid.
According to the Pythagorean theorem, we have:
x^2 + b^2 = c^2
where x is the length of the unknown leg, b is the length of the base of the square pyramid, and c is the slant height.
Substituting the given values, we get:
x^2 + b^2 = 25^2
x^2 + b^2 = 625
Now, we also know that the height of the pyramid is 20 miles. The height forms a right triangle with the base and the slant height as the hypotenuse. Therefore, we can use the Pythagorean theorem again:
b^2 + 20^2 = c^2
b^2 + 400 = 625
b^2 = 625 - 400
b^2 = 225
Now, we can substitute this value of b^2 back into the first equation:
x^2 + 225 = 625
x^2 = 625 - 225
x^2 = 400
Taking the square root of both sides:
x = sqrt(400)
x = 20
Therefore, the length of the base of the square pyramid is 20 miles.
To find the base length of the square pyramid, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse (the longest side).
In this case, we can consider the unknown leg as one of the legs of a right triangle, with the slant height as the hypotenuse and the base length as the other leg.
Let's denote the base length as 'x'.
Using the Pythagorean theorem, we can write the equation as follows:
x^2 + (20^2) = (25^2)
Simplifying the equation, we get:
x^2 + 400 = 625
Subtracting 400 from both sides, we get:
x^2 = 225
Taking the square root of both sides, we get:
x = 15
Therefore, the base length of the square pyramid is 15 miles long.