A right triangle has legs a and b and hypothenuse c. Prove that the altitude h to the
hypothenuse is h = a/b
To prove that the altitude h to the hypotenuse is h = a/b, we can use the concept of similar triangles.
Let's consider the right triangle with legs a and b and hypotenuse c. If we drop a perpendicular from the right angle to the hypotenuse, we create two smaller right triangles within the original triangle.
Let the altitude h split the hypotenuse into two segments, x and y. We can then write the following relationships:
In the smaller right triangle with legs h and x, we have:
(h/a) = (b/c) ... (1)
In the smaller right triangle with legs h and y, we have:
(h/b) = (a/c) ... (2)
Adding equations (1) and (2) together, we get:
(h/a) + (h/b) = (b/c) + (a/c)
h(1/a + 1/b) = (b + a) / c
h = (h / c) * (b + a) ... (3)
Now, let's consider the larger right triangle with legs a and b:
By the Pythagorean theorem, we have:
a^2 + b^2 = c^2
Rearranging this equation, we get:
c = (a^2 + b^2)^(1/2)
Substitute c = (a^2 + b^2)^(1/2) into equation (3), we get:
h = (h / (a^2 + b^2)^(1/2)) * (a + b)
Simplifying the expression, we get:
h = a * b / (a^2 + b^2)^(1/2)
Since c = (a^2 + b^2)^(1/2), we can rewrite the expression as:
h = a * b / c
Therefore, we have proven that the altitude h to the hypotenuse is h = a/b.