Given: ~ with altitudes CR and ZS.
ZS = 12, XY = 15, YZ = 18 and AB = 20, find the measure of CR.
(2 points)
A) 25
B) 9
C) 17
D) 16
To find the measure of CR, we can use the theorem that states that in a right triangle, the product of the two segments of one altitude is equal to the product of the two segments of the other altitude.
In this case, we have altitudes CR and ZS. The lengths of these altitudes are not given, but we are given the lengths of the sides of the triangle. We can use these side lengths to find the lengths of the altitudes.
First, let's find the lengths of the sides of the triangle using the given information:
- XY = 15
- YZ = 18
- AB = 20
We can see that the triangle XYZ is a right triangle because AB, which is the hypotenuse, is the longest side.
Now, let's find the lengths of CR and ZS.
To find the length of ZS, we will use the fact that the product of the two segments of one altitude is equal to the product of the two segments of the other altitude:
ZS * ZY = CR * CA
We know the length of ZS is 12, and ZY is the length of the altitude from vertex Y to line ZX. We can find the length of ZY using the Pythagorean theorem:
ZY^2 = XY^2 - ZX^2
= 15^2 - (18^2 - 12^2)
= 225 - (324 - 144)
= 225 - 180
= 45
ZY = √45
= 3√5
Now, we can substitute the values into the equation and solve for CR:
12 * (3√5) = CR * 20
36√5 = 20CR
CR = (36√5) / 20
= 9√5 / 5
= 9 * (√5 / 5)
= 9 * (√5 / √25)
= 9 * (√5 / 5)
= 9/5 * (√5)
So, the measure of CR is 9√5/5.
None of the answer choices given match this exact value, so we can conclude that none of the given answer choices are correct.