Question

the vertex angle of an isosceles triangle is three times the measure of either base angle. find the area if the measure of the congruent sides is 14 in.

I know the answer is 176.2 cm squared, but I have no idea how to get there!

Answer 1

We are given this:

Answer 2

So far so good?

Answer 3

Yes.

Answer 4

To find rhe area of a triangle, we need a base and a height.
I'll draw the height in our triangle below.

Answer 5

We need to find h, the height.
Using trig, and knowing the side of 14 in., we can find h if we first find the measure of the angles x.

Answer 6

Step 1. Let's find the measure of the base angles, x.
What do you know about the sum of the measures of the interior angles of a triangle?

Answer 7

They add up to 180°. Right?

Answer 8

Correct.
We can write an equation and solve for x:
3x + x + x = 180
Can you solve for x?

Answer 9

5x = 180
5x/5 = 180/5
x = 36
The base angles measure 36 deg.

Answer 10

5x=180
X=36

Answer 11

Now look at the right triangle on the left side.
Its hypotenuse measures 14 in.
One acute angle measures 36 deg.
Using trig of right triangles, we can find h.
For the angle of 36 deg, h is the opposite leg.
Which trig function relates the opposite leg and the hypotenuse?

Answer 12

Do you know SOHCAHTOA?

Answer 13

I'm a 9th grade student in geometry, so we haven't covered trig very much. But yes, sin?

Answer 14

Great. You know your stuff.

\(\sin A = \dfrac{opp}{hyp} \)

Answer 15

We can use the info we have in the sin equation:

\(\sin 36^o = \dfrac{h}{14} \)

Now we solve for h:

\(h = 14 \sin 36^o\)

Answer 16

Step 2. Now we need a base.
We use trig again.

We now have the adjacent leg and the hypotenuse.
Using SOHCAHTOA which trig function relates the adjacent leg to the hypotenuse?