# A triangle has corners at points A, B, and C. Side AB has a length of #36 #. The distance between the intersection of point A's angle bisector with side BC and point B is #8 #. If side AC has a length of #42 #, what is the length of side BC?

Let the intersection point be D (see diagram)

Using Laws of sine :

a) triangle ABD

b) triangle ADC

Let E be the distance between D and C

Recall that

Eq (1) = Eq (2)

Length of BC

By signing up, you agree to our Terms of Service and Privacy Policy

To find the length of side BC, we can use the angle bisector theorem.

Given: AB = 36 AC = 42 Distance from the intersection of point A's angle bisector with side BC to point B = 8

Let's denote the length of side BC as x.

According to the angle bisector theorem, the ratio of the lengths of the two segments of a side bisected by an angle bisector is equal to the ratio of the lengths of the other two sides of the triangle. Mathematically, this can be represented as:

[ \frac{AB}{AC} = \frac{BD}{DC} ]

Where BD is the length from point B to the intersection of the angle bisector with side BC, and DC is the length from that intersection to point C.

Given: AB = 36 AC = 42 BD = 8 DC = x - 8 (because the whole length BC is x)

Plugging in the given values:

[ \frac{36}{42} = \frac{8}{x - 8} ]

Simplifying:

[ \frac{6}{7} = \frac{8}{x - 8} ]

Cross-multiplying:

[ 6(x - 8) = 7(8) ]

Expanding:

[ 6x - 48 = 56 ]

Adding 48 to both sides:

[ 6x = 104 ]

Dividing both sides by 6:

[ x = \frac{104}{6} ]

[ x = 17.33 ]

So, the length of side BC is approximately 17.33 units.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A triangle has corners at points A, B, and C. Side AB has a length of #16 #. The distance between the intersection of point A's angle bisector with side BC and point B is #12 #. If side AC has a length of #8 #, what is the length of side BC?
- Enter the proportional segment lengths into the boxes to verify that ¯¯¯QS¯∥MN¯ . ___ /1.5= ___ / ___?
- A triangle has corners at points A, B, and C. Side AB has a length of #27 #. The distance between the intersection of point A's angle bisector with side BC and point B is #18 #. If side AC has a length of #18 #, what is the length of side BC?
- Triangle A has an area of #4 # and two sides of lengths #9 # and #3 #. Triangle B is similar to triangle A and has a side with a length of #32 #. What are the maximum and minimum possible areas of triangle B?
- Triangle A has sides of lengths #2 ,3 #, and #4 #. Triangle B is similar to triangle A and has a side of length #5 #. What are the possible lengths of the other two sides of triangle B?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7