SOLUTION: Problem 51: http://www.newton.k12.oh.us/Downloads/Section%205-5%20HW.pdf Solve the triangle. (Hint: Draw the altitude from Y) Given: The triangle is not a right triangle. It ha

Algebra ->  Triangles -> SOLUTION: Problem 51: http://www.newton.k12.oh.us/Downloads/Section%205-5%20HW.pdf Solve the triangle. (Hint: Draw the altitude from Y) Given: The triangle is not a right triangle. It ha      Log On


   



Question 1004205: Problem 51: http://www.newton.k12.oh.us/Downloads/Section%205-5%20HW.pdf
Solve the triangle. (Hint: Draw the altitude from Y)
Given:
The triangle is not a right triangle.
It has points X, Y and Z.
The base is y.
XY is 16, YZ is 24, and XZ is y.
Angle X is 30 degrees.
What I've solved so far:
Angle Y is 120 degrees
Angle Z is 30 degrees.
Is this correct? It does not seem so...
Thank you.


Found 2 solutions by josgarithmetic, MathLover1:
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
I spent a few minutes checking the drawing in your book and reading the brief instructions. DRAW the altitude. Doing so will give you two right triangles which share this altitude. Keep working through and you may find expressions whose sum would be y.

Below is a precise description with some calculation, but not taken to completion. You should be able to do so with very little trouble.

---
Part of the work...
Using the right triangle on the left side, let h be the height or altitude for BOTH triangles.
h%2F16=sin%2830%29
h=16%2Asin%2830%29

The segment length y, the base for the triangle (before or after making the altitude) is composed of two parts. y%5B1%5D and y%5B2%5D; sub 1 for the part in the left right triangle, sub 2 for the one on the right.
There will be y=y%5B1%5D%2By%5B2%5D.

Left side right triangle, according to pythagorean theorem formula,
%28y%5B1%5D%29%5E2%2B%2816%2Asin%2830%29%29%5E2=16

The right side right triangle, gives you %28y%5B2%5D%29%5E2%2B%2816%2Asin%2830%29%29%5E2=24%5E2.

What to do from here, solve the separate sub-parts for y%5B1%5D and y%5B2%5D, and sum them. y=y%5B1%5D%2By%5B2%5D

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

use hint "draw the altitude from y"
let the altitude be x, then from first right angle triangle we have
sin%2830%29=x%2F16
1%2F2=x%2F16
%281%2F2%2916=x
x=8
let angle at Z be theta
sin%28theta%29=x%2F24....since x=8
sin%28theta%29=8%2F24
sin%28theta%29=1%2F3
theta=19.5

since angle Y is divided in two angles by altitude, let them be alpha and beta,
so,Y=alpha%2Bbeta
alpha is in first right angle triangle where is angle of 30 degrees, so alpha=60 degrees
in second right angle triangle, one angle is theta=19.5, and beta=70.5
now we have angle Y=60%2B70.5=>Y=130.5
side y is divided by altitude into a and b
since we know all angles and two sides in each right angle triangle, we can find a and b
cos%2830%29=a%2F16
a=sqrt%283%29%2F2%2A16
a=sqrt%283%29%2A8
a=13.85640646055102
a=13.9

cos%2819.5%29=b%2F24
b=0.795815%2A24
b=22.6

y=a%2Bb
y=13.9%2B22.6
highlight%28y=36.5%29