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?

Question

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?

Jayden Jackson · Accepted Answer

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

Carson Arnold · Answer

undefined 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.