15. The altitude of the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 4 cm. and 9 cm. What is the length of the altitude?
altitude h
side beside 4= a
side beside 9 = b
h^2 + 16 = a^2
h^2 + 81 = b^2
-------------------- add
2h^2 + 97 = a^2 + b^2 = 13^2 = 169
2 h^2 = 72
h = 6
using similar triangles,
4/x = x/9
x^2 = 36
x = 6
To solve this problem, we need to use the concept of similar triangles.
Let's consider a right triangle with sides labeled as follows:
- Hypotenuse: C
- Altitude: h
- Legs: a and b
We are given that the altitude divides the hypotenuse into segments of lengths 4 cm and 9 cm. So, we can say:
AC = 4 cm
BC = 9 cm
Based on the concept of similar triangles, we know that the two smaller triangles formed by the altitude are similar to the larger right triangle.
Using this information, we can set up the following proportion:
AC / C = h / BC
Substituting the given values, we have:
4 cm / C = h / 9 cm
To solve for h, we need to solve this proportion for C.
We can rearrange the equation by cross-multiplying:
4 cm * BC = C * h
Substituting the given values:
4 cm * 9 cm = C * h
Simplifying the equation:
36 cm² = C * h
Now, we need to find the value of C. We can do this by using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have:
C² = a² + b²
Since we are given the lengths of the segments, we can write:
C² = 4 cm² + 9 cm²
Simplifying the equation:
C² = 16 cm² + 81 cm²
C² = 97 cm²
Taking the square root of both sides, we find:
C = √97 cm
Now that we have the value of C, we can substitute it back into the equation:
36 cm² = (√97 cm) * h
To isolate h, we divide both sides by √97 cm:
h = 36 cm² / (√97 cm)
Using a calculator, we can find an approximate value for h to be:
h ≈ 12.6 cm
Therefore, the length of the altitude is approximately 12.6 cm.