A triangle has corners at points A, B, and C. Side AB has a length of #32 #. The distance between the intersection of point A's angle bisector with side BC and point B is #16 #. If side AC has a length of #28 #, what is the length of side BC?

Answer 1

Jayden Jackson

30

Firstly , let the point where the angle bisector intersects with side BC be D.

Then by the #color(blue)" Angle bisector theorem " #

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

Require to find DC.

Substitute the appropriate values into the ratio to obtain.

#(16)/(DC) = 32/28 #

now cross-multiply : #32xxDC = 28xx16 #

To obtain DC , divide both sides by 32

#( cancel(32) DC)/cancel(32) = (28xx16)/32 #

# rArr DC = 14 #

Now , BC = BD + DC = 16 + 14 = 30

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Answer 2

Carson Arnold

Using the Angle Bisector Theorem, we can determine that the length of side BC can be found using the proportion:

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

Given that AC = 28 and AB = 32, and letting x represent the length of BC, we can set up the proportion:

[\frac{28}{32} = \frac{28-x}{16}]

Solving this equation will give us the length of BC:

[28 \times 16 = 32 \times (28-x)] [448 = 896 - 32x] [32x = 896 - 448] [32x = 448] [x = \frac{448}{32}] [x = 14]

So, the length of side BC is 14 units.

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Answer from HIX Tutor

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