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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 #8 #. If side AC has a length of #32 #, what is the length of side BC?

Answer 1

Mason Adams

Length of side BC = 24

Let the point where the angle bisector intersects with side BC be D

#"using the "color(blue)"angle bisector theorem"#

#(AB)/(AC)=(BD)/(DC)#

#16 / 32 = 8 / (DC)#

#DC = (8*32) / 16 = 16#

#BC = BD+DC= 8+16 =24#

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Answer 2

Sadie Adams

Using the Angle Bisector Theorem, we can find the length of side BC. The theorem states that in a triangle, the ratio of the lengths of the two segments of a side bisected by an angle bisector is equal to the ratio of the other two sides of the triangle.

Let D be the intersection of the angle bisector of angle A with side BC. According to the theorem:

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

Given that AB = 16 and AC = 32:

[\frac{BD}{DC} = \frac{16}{32} = \frac{1}{2}]

Since BD = 8 (given in the question), we can find DC:

[8 = \frac{1}{2} \times DC]

[DC = 16]

Therefore, BC = BD + DC = 8 + 16 = 24.

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Answer from HIX Tutor

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.