Home > Geometry > Angle Bisector Theorem

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 #4 #. If side AC has a length of #28 #, what is the length of side BC?

Answer 1

Maverick Smith

#norm(B-C)= 7.5#

Referencing the attached figure, we have two triangles

#ABD# and #ADC#. Aplying the sinus law we have

#ABD-> sin(alpha)/4=sin(beta)/32#
#ADC->sin(alpha)/x = sin(gamma)/28#

but #beta = pi-gamma# so the first equation reads

#ABD-> sin(alpha)/4=sin(gamma)/32# because #sin(pi-gamma)=sin(gamma)#

Dividing term to term the two equations we have

#4/x=32/28# giving #x = 7/2# so #norm(B-C)=4+7/2 = 7.5#

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

Answer 2

Wyatt Anderson

Using the Angle Bisector Theorem, the length of side BC can be found using the formula:

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

Given: [ AB = 32 ] [ AC = 28 ] [ BD = 4 ]

Substituting the given values: [ \frac{32}{28} = \frac{4}{DC} ]

Solving for DC: [ DC = \frac{28 \times 4}{32} = 3.5 ]

So, the length of side BC is 3.5.

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

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.