The altitude of the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 14 and 8. What is the length of the altitude?
a^2 + b^2 = c^2
let a = altitude
b = 8
c = 14
a^2 + 8^2 = 14^2
a^2 = 196 - 64
a^2 = 132
a = 11.49
To find the length of the altitude of the hypotenuse in a right triangle, we can use the concept of similar triangles.
Let's denote the length of the altitude as "x". According to the problem, the altitude divides the hypotenuse into segments of lengths 14 and 8.
Using the concept of similar triangles, we can set up the following proportion:
14/x = x/8
To solve for "x", we can cross-multiply:
14 * 8 = x * x
112 = x^2
Taking the square root of both sides:
√112 = √(x^2)
√112 = x
Finally, simplifying the square root, we find:
x ≈ 10.58
So, the length of the altitude of the hypotenuse is approximately 10.58.
To find the length of the altitude of the right triangle, you can use the Pythagorean Theorem. The theorem states that in a right 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.
Let's denote the length of the altitude as "h", the length of one of the legs of the triangle as "8", and the length of the other leg as "14". According to the given information, the altitude divides the hypotenuse into segments of lengths 14 and 8.
We can form two right triangles using the altitude as the height in both cases. One triangle has a base of 14 and height "h", and the other triangle has a base of 8 and height "h". Using the Pythagorean theorem in both cases, we get:
For the first triangle:
14^2 = h^2 + 8^2
196 = h^2 + 64
h^2 = 196 - 64
h^2 = 132
For the second triangle:
8^2 = h^2 + 14^2
64 = h^2 + 196
h^2 = 64 - 196
h^2 = -132
Since we are dealing with lengths, we can disregard the negative value of h². Therefore, the length of the altitude, h, is equal to the square root of 132:
h = sqrt(132)
Using a calculator, we can find that the length of the altitude is approximately 11.49 units.