A triangle has corners at points A, B, and C. Side AB has a length of #12 #. The distance between the intersection of point A's angle bisector with side BC and point B is #7 #. If side AC has a length of #21 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #12  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #7  #.  If side AC has a length of #21  #, what is the length of side BC?

Claire Akin · Accepted Answer

BC=19.25

Here is a sketch (not to scale)

To calculate the length of BC ,we require to find the length of DC.
This can be done by using the #color(blue)" Angle bisector theorem"#

For triangle ABC this is.

#color(red)(|bar(ul(color(white)(a/a)color(black)((BD)/(DC)=(AB)/(AC))color(white)(a/a)|)))#
Substitute the appropriate values into the ratio.

#rArr7/(DC)=12/21#
now cross-multiply

#rArrDCxx12=7xx21rArrDC=(7xx21)/12=12.25#
Thus BC = BD + DC = 7 + 12.25 = 19.25

Autumn Armstrong · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC. The theorem states that in a triangle, if a line bisects an angle, it divides the opposite side into segments that are proportional to the other two sides.

Let D be the point where the angle bisector of angle A intersects side BC.

According to the theorem, $ \frac{{BD}}{{DC}} = \frac{{AB}}{{AC}} $.

Given that AB = 12 and AC = 21, we can find the ratio BD:DC.

$ \frac{{BD}}{{DC}} = \frac{{12}}{{21}} $.

Simplifying, we get $ \frac{{BD}}{{DC}} = \frac{{4}}{{7}} $.

Let x be the length of BC. Then BD + DC = x.

Since BD:DC = 4:7, BD = $ \frac{{4x}}{{11}} $ and DC = $ \frac{{7x}}{{11}} $.

We know that BD + DC = x. Substituting the expressions for BD and DC, we get:

$ \frac{{4x}}{{11}} + \frac{{7x}}{{11}} = x $.

Solving for x:

$ \frac{{11x}}{{11}} = x $.

Thus, x = 11.

Therefore, the length of side BC is 11.