Question

3. In the diagram below of right triangle ACB, altituDB find the length of CD in simplest radical form4,

Answer 1

ACB, ACD and DCB are all right triangles. We can then represent the relationships of their legs through the Pythagorean theorem. 

For triangle ACB:
7^2=(AC)^2+(BC)^2     (1)

For triangle ACD:
(AC)^2=3^2+(CD)^2     (2)

For triangle DCB:
(BC)^2=(CD)^2+4^2     (3)

To make things simpler, we let
(AC)=x
(BC)=y
(CD)=z

Hence we have:
49=x^2+y^2     (1)
y^2=z^2+16     (2)
x^2=9+z^2       (3)

To solve for the unknown c, let us first substitute equation (2) in (1)
49=x^2+z^2+16
33=x^2+z^2     (4)

Then, we can substitute equation (3) in (4)
33=9+z^2+z^2
24=2z^2
z^2=12
z=2sqrt(3)

Thus, the length of z or (CD) is 2sqrt(3) in simplest radical form.