9. Find the length of the altitude drawn to the hypotenuse.
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28
7.5
14
9
To find the length of the altitude drawn to the hypotenuse, we can use the formula for the area of a triangle: A = 1/2 * base * height.
In this case, the hypotenuse is the base of the triangle and the altitude is the height. Let's call the length of the hypotenuse "c" and the length of the altitude "h".
Since it's a right triangle, we can use the Pythagorean theorem to relate the lengths of the two legs (a and b) and the hypotenuse (c):
a^2 + b^2 = c^2
Let's assume that one leg of the triangle is "a" and the other leg is "b". Since the altitude splits the right triangle into two smaller triangles, the lengths of the legs of those triangles are "h" and "c-h", respectively.
Using the Pythagorean theorem for the left triangle:
h^2 + a^2 = c^2
Using the Pythagorean theorem for the right triangle:
(c-h)^2 + b^2 = c^2
We can now solve these two equations simultaneously to find the lengths of the legs in terms of "c" and "h".
From the first equation:
h^2 + a^2 = c^2
a^2 = c^2 - h^2
a = sqrt(c^2 - h^2)
From the second equation:
(c-h)^2 + b^2 = c^2
b^2 = c^2 - (c-h)^2
b^2 = c^2 - (c^2 - 2ch + h^2)
b^2 = 2ch - h^2
b = sqrt(2ch - h^2)
Now, let's substitute these expressions for "a" and "b" into the formula for the area of a triangle:
A = 1/2 * base * height
A = 1/2 * c * h
A = ch/2
But we also know that the area of a triangle can be calculated as 1/2 * base * height. In this case, the base is the length of the hypotenuse "c" and the height is the length of the altitude "h".
Therefore, we have:
A = 1/2 * c * h
A = ch/2
Setting these two expressions for the area equal to each other, we can solve for "h":
28 = ch/2
h = 2*28/c
h = 56/c
So, the length of the altitude drawn to the hypotenuse is 56/c.