In triangle \(\triangle ABC,\) we have that \(AB = AC = 14\) and \(BC = 26.\) What is the length of the shortest angle bisector in \(ABC\)? Express your answer in simplest radical form.

VooFIX Nov 15, 2019

#1**+2 **

It forms an isoceles triangle, because two sides are congruent (both are 14).

We have two choices, either the base angles or the vertex angle.

The only thing I can think of right now is to use coordinate geometry.... I don't know? Help!

CalculatorUser Nov 15, 2019

#2**+1 **

Wait since its isosceles then the base should be 13 and the hypotenuse is 14 and therefore we can solve the height. In Isosceles, the angle bisector is the perpendicular bisector, pretty sure.

VooFIX Nov 15, 2019

#3**+1 **

yeah... I dont knwo where to continue.

The hight is going to be \(3\sqrt{3}\)

CalculatorUser
Nov 15, 2019

#4**+1 **

be careful with these!!

ok let's see if i can walk you through this bc i literally just finished the geometry unit:

so you have triangle ABC

and like CU said since they GIVE us the sidelength of AB and AC as 14 (which are equal) then it means that our triangle is isoceles. since we also know that BC is 26.

So! do you remember how to find the angle bisector or shall we work through that together :)?

Nirvana
Nov 15, 2019

#5**0 **

nope I havent learn this yet, I just know that the angle is divided into TWO lollololol

CalculatorUser
Nov 15, 2019

#6**0 **

yes great job CU! but you're missing just a tiny thing ........the angle bisector MUST divide the angle into two CONGRUENT angles!!

Perpendicular Bisector: given two points you must find the slope of the segment and find the perpendicular slope/equation of the perpendicular bisector

Angle Bisector: divides angles into two congruent angles (shoot lol i accidentally said sides)

Voofix! whenever you see bisector immediately think:

divides into two equal_________

Nirvana
Nov 15, 2019

#9**0 **

but if we know that the angles are divided into two congruent parts, how do we solve knowing the sides.

I mean, I try to refrain from cheating with Trigonometry, so below is the info I gathered.

Since the angle is bisected, we have

4x + y = 180.

Y is the vertex angle, and x is one of the base angles that are bisected, so there are total 4 of them.

I cannot find any special triangles anywhere....

CalculatorUser
Nov 15, 2019

#7**+1 **

The shortest angle bisector will be drawn from the greatest angle - angle BAC - to side BC

Since the triangle is isosceles, this angle bisector will = the altitude drawn from BC to A

We can use the Pythagorean Theorem to find this

Call the bisector AD and we have that

AD = √ [ AB^2 - (BC/2)^2 ]

AD = √ [ 14^2 - 13^2]

AD = √[196 - 169 ]

AD = √27 = 3√3

Here's a pic :

CPhill Nov 15, 2019

#10**+1 **

Yup I understand,

but I don't understand how do you know the angle bisector of BAC is going to be the shortest segment...

Is there a theorem on this?

Wait Oh I think we learned in class that the smaller the angle, the LONGER the segment.

So the BIGGER the angle, the SMALLER the segment.

CalculatorUser
Nov 15, 2019