Question

"Find y". I got this wrong and want to understand why and how to do the problem. Can someone please help?

Answer 1

Answer:

$y=\frac{15\sqrt{3} }{4}$

Step-by-step explanation:

Well what you have here are two right triangles within another right triangles created by an altitude. This results in 3 similar triangles (see attached for reference)

As a result , using the properties of similar triangles, and taking ratios of their sides of the medium triangle and the largest triangle, we can form the following relationship:

$\frac{b}{z}=\frac{y}{x} = \frac{z}{a+b}$  (we know that a+b = 15), hence the equation becomes:

$\frac{b}{z}=\frac{y}{x} = \frac{z}{15}$  ----------eq 1

If we consider the largest triangle, we will see that

cos 30° = z / 15

but from our special angles (see second attachment) we know that cos 30° = (√3)/2

hence

(√3)/2 = z / 15

z = (15√3) / 2 --------eq 2

from the same largest triangle, we will also see that

sin 30° = x / 15

similarly from our special angles, we know that sin 30° = 1/2

hence,

1/2 = x/15

x = 15/2 --------eq 3

now that we have values for x and z, we can neglect the first term of equation 1 and form an equality with x, y and z

$\frac{y}{x} = \frac{z}{15}$

$y = x\frac{z}{15}$  (substituting the values for x and z that we found above)

y = (15/2) · [ (15√3) / 2] / 15

y = (15√3) / 4

$y=\frac{15\sqrt{3} }{4}$