# Thread: [SOLVED] math b - triangles

1. ## [SOLVED] math b - triangles

hi I'm new to the forum and not sure where to post. I just have a few questions that i need help answering.

-Find the indicated length for each equilateral triangle with given lengths..

altitude 3 square root 3; side? and side 12; altitude ?

-A boy 6 feet tall casts a shadow 15 feet long. at the same time a tree casts a shadow of length 50 feet. what is the height, in feet, of the tree?

-The three sides of a triangle measure 4, 8 and 9. find the length of the longest side of a similar triangle whose perimeter is 63.

2. Originally Posted by ThebrightOne
hi I'm new to the forum and not sure where to post. I just have a few questions that i need help answering.

-Find the indicated length for each equilateral triangle with given lengths..

altitude 3 square root 3; side? and side 12; altitude ?

-A boy 6 feet tall casts a shadow 15 feet long. at the same time a tree casts a shadow of length 50 feet. what is the height, in feet, of the tree?

-The three sides of a triangle measure 4, 8 and 9. find the length of the longest side of a similar triangle whose perimeter is 63.

I've attached images for questions 1, 2, and 3

1. In an equilateral triangle the altitude bisects a side (divides it in 2), so if $\displaystyle x$ is the length of the side, by pythagoras we have:

$\displaystyle (3\sqrt{3})^2 + \left(\frac{x}{2}\right)^2 = x^2$

$\displaystyle 27+\frac{x^2}{4}=x^2$

$\displaystyle 108 + x^2 = 4x^2$

$\displaystyle 3x^2 = 108$

$\displaystyle x^2= 36$

$\displaystyle x = 6$ (only the positive root because length is positive)

2. Let a be the altitude:

$\displaystyle a^2+\left(\frac{12}{2}\right)^2 =12^2$

$\displaystyle a^2 + 36 = 144$

$\displaystyle a^2 = 108$

$\displaystyle a = \sqrt{108}=6\sqrt{3}$

3. Let x be the height of the tree. Using similar triangles,

$\displaystyle x:6 = 50:15$

$\displaystyle \frac{x}{6}=\frac{50}{15}$

$\displaystyle x = 20$

4. If the triangle is similar, then the sides must be in proportion with each other, but they can all be enlarged by a factor $\displaystyle k$.

So by equating the sum of the sides of the enlarged triangle with its perimeter:

$\displaystyle 4k + 8k + 9k = 63$

$\displaystyle k = 3$

So the factor is 3, and hence the largest side is $\displaystyle 9k = 9(3) = 27$