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### Course: High school geometry > Unit 4

Lesson 4: Angle bisector theorem# Using the angle bisector theorem

Sal uses the angle bisector theorem to solve for sides of a triangle. Created by Sal Khan.

## Want to join the conversation?

- why cant you just use the pythagorean theorem to find the side that x is on and then subtract the half that you know?(32 votes)
- The pythagorean theorem only works on right triangles, and none of these triangles are shown to have right angles, so you can't use the pythagorean theorem.(65 votes)

- At0:40couldnt he also write 3/6 = 2/x or 6/3 = x/2? Thanks(11 votes)
- Yes, you could. switching the denominator and the numerator on both sides of an equation has no effect on the result. An example: If you have 3/6 = 3/6. Switch the denominator and numerator, and get 6/3 = 6/3. That is the same thing with x. 6/3 = x/2 can be 3/6 = 2/x. Now, when using the Angle Bisector theorem, you can also use what you just did. You will get the same result! Hope this answers your question.(12 votes)

- What's the purpose/definition or use of the Angle Bisector Theorem? Could someone please explain this concept to me?(8 votes)
- In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC:

http://en.wikipedia.org/wiki/Angle_bisector_theorem(11 votes)

- this may not be a mistake but when i did this in the questions it said i had got it wrong so clicked hints and it told me to do it differently to how Sal khan said to do it. is there a way of telling which one to use or have i missed something?(6 votes)
- That sort of thing has happened to me before. If you learn more than one correct way to solve a problem, you can decide which way you like best and stick with that one. I've learned math problems that required doing DOZENS of practice problems because I'd get all but the last one right over and over again. Study the hints or rewatch videos as needed. Keep trying and you'll eventually understand it. ;)(11 votes)

- I'm still confused, why does this work?(7 votes)
- See an explanation in the previous video, Intro to angle bisector theorem: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-angle-bisector-theorem/v/angle-bisector-theorem-proof(0 votes)

- I found the answer to these problems by using the inverse function like:

sin-1(3/4) = angleº

and got the correct answers but I know that these inverse functions only work for right triangles... can someone explain why this worked?(3 votes)- The trig functions work for any angles. The right triangle is just a tool to teach how the values are calculated.(2 votes)

- not for this specifically but why don't the closed captions stay where you put them? They sometimes get in the way. The videos didn't used to do this.(3 votes)
- What is the angle bisector theorem?.(1 vote)
- In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.(3 votes)

- Who invented math? I can't do math very well.(1 vote)
- No one INVENTED math, more like DISCOVERED it. Math is really just facts, so you can't invent facts.(3 votes)

- who is sal talking about when he says 'they'?(1 vote)

## Video transcript

I thought I would do a few
examples using the angle bisector theorem. So in this first
triangle right over here, we're given that this
side has length 3, this side has length 6. And this little
dotted line here, this is clearly
the angle bisector, because they're telling
us that this angle is congruent to that
angle right over there. And then they tell
us that the length of just this part of this
side right over here is 2. So from here to here is 2. And that this length is x. So let's figure out what x is. So the angle bisector
theorem tells us that the ratio of 3 to 2 is
going to be equal to 6 to x. And then we can
just solve for x. So 3 to 2 is going to
be equal to 6 to x. And then once again, you
could just cross multiply, or you could multiply
both sides by 2 and x. That kind of gives
you the same result. If you cross multiply, you get
3x is equal to 2 times 6 is 12. x is equal to, divide both
sides by 3, x is equal to 4. So in this case,
x is equal to 4. And this is kind of interesting,
because we just realized now that this side, this entire
side right over here, is going to be equal to 6. So even though it
doesn't look that way based on how it's drawn, this is
actually an isosceles triangle that has a 6 and a 6, and then
the base right over here is 3. It's kind of interesting. Over here we're given that this
length is 5, this length is 7, this entire side is 10. And then we have this angle
bisector right over there. And we need to figure out just
this part of the triangle, between this point, if
we call this point A, and this point right over here. We need to find the length
of AB right over here. So once again, angle bisector
theorem, the ratio of 5 to this, let me do this in a
new color, the ratio of 5 to x is going to be equal
to the ratio of 7 to this distance
right over here. And what is that distance? Well, if the whole thing
is 10, and this is x, then this distance right over
here is going to be 10 minus x. So the ratio of 5 to x is
equal to 7 over 10 minus x. And we can cross
multiply 5 times 10 minus x is 50 minus 5x. And then x times
7 is equal to 7x. Add 5x to both sides
of this equation, you get 50 is equal to 12x. We can divide both sides by
12, and we get 50 over 12 is equal to x. And we can reduce this. Let's see if you divide the
numerator and denominator by 2, you get this is the
same thing as 25 over 6, which is the same thing, if
we want to write it as a mixed number, as 4, 24
over 6 is 4, and then you have 1/6 left over. 4 and 1/6. So this length right
over here is going, oh sorry, this length right
over here, x is 4 and 1/6. And then this
length over here is going to be 10 minus 4 and 1/6. What is that? 5 and 5/6.