Question

Find the length of the altitude of an isosceles triangle with vertex angle 120 degrees and base length of 30 centimeters. Give answer in simplified radical form. Draw a diagram first.

@mtbender74

Answer 1

oh...forgot the base of 30 cm in the drawing

Answer 2

if angle B is 120, how much do angles A and C measure?

Answer 3

It's okay, I know where it goes.

Answer 4

30

Answer 5

good...so let's use that.

I've also included the altitude BD...

Answer 6

Do you know what the altitude BD does to side AC? (since the triangle is isosceles)

Answer 7

No..

Answer 8

fair enough :) The vertex angle altitude draw to the base bisects the base equally.

So BD cuts the base in half. we can prove this with AAS congruence, but better to just take my word for it :P

that being said, the measure of AD and DC are therefore...?

Answer 9

15!

Answer 10

exactly! :)

So now we need another fun fact to put in our toolbox...good thing it's a large toolbox because we've got a lot of tools by now.

You see my avatar? it's called a unit circle. it tells us the side lengths of several special triangles. Not the same special triangles from before (3-4-5, and 5-12-13), but rather special triangles with certain angle measures.

Answer 11

The first, it what's called a 45-45-90 triangle. It is an isosceles right triangle. the sides will always be in the ratio of 1:1:sqrt(2)

So, if you know one leg of a 45-45-90 triangle is, let's say, 5, then you know the triangle measures are 5, 5, and 5sqrt(2)

Answer 12

you follow?

Answer 13

Yup

Answer 14

good. now with the 45-45-90, it's easy to remember how the measures relate to the sides because the two legs always get the same numbers and the hypotenuse gets the longer one...

but let's now switch to the other one...the 30-60-90 triangle. For this guy, the ratio is 1:2:sqrt(3). but there's a catch...

the 45-45-90 ratio was listed leg:leg:hypotenuse (1:1:sqrt(2))
the 30-60-90 ratio is listed short leg:hypotenuse:long leg (1:2:sqrt(3))

it's easy to screw that up so i'm warning you now.. :)

Answer 15

stop me if you know all of this... :)

Answer 16

So the answers are 15-15-15sqrt(2) and 15-30-15sqrt(3)?

Answer 17

And that would make BD=15

Answer 18

not quite...you're close...but first, which special triangle do we have? 45-45-90? or 30-60-90?

Answer 19

30-60-90 for ABD and BDC right?

Answer 20

correct. so...we need to consider which side we have the value for so we can get the ratio right.