The perimeter of triangle ABC is 120. LIne segment BD bisects <B. Line segment AD= 15, and line segment DC= 21. FInd AB and BC.

Thanks.

This is a very tough question for grade 8, I am totally impressed.

make your sketch, let
AB = a
BC = b
BD = x
Angle B = 2Ø , so that angle ABD = Ø and angle BCD = Ø

I am going to use the fact that if we have triangle with sides a and b with Ø the angle between them
the area = (1/2)ab sinØ

area of triangle ABD= (1/2)ax sinØ
area of triangle BCD=(1/2)bx sinØ

But if we consider the bases along AC , the both have the same measured from B
so they are in the ratio of 15:21 or 5:7

then:
(1/2)axsinØ/( (1/2)bxsinØ ) = 5/7
canceling reduces this to
a/b = 5/7
5b = 7a
b = 7a/5 = 1.4a ---- ******

now let's use the perimeter:
a + b + 15+21 = 120
a+b =84 -----******

sub in b = 1.4a into the above equation
a+b = 84
a + 1.4a = 84
2.4a = 84
a = 84/2.4 = 35
then b = 1.4(35) = 49


great question !!!!

or, using the angle bisector theorem,

AD/DC = AB/BC
so, as Reiny calculated,

AB/BC = 5/7
AD+BC = 84
5x+12x=84
x=7

so, AB = 35
BC = 49

oops

5x+7x=84
x=7

Thanks.

To find the lengths of AB and BC, we can apply the triangle angle bisector theorem which states that if a line segment divides two sides of a triangle proportionally, then it is an angle bisector.

Let's first look at triangle ABC. The perimeter of triangle ABC is given as 120.

Perimeter of triangle ABC = AB + BC + AC.

But we are given that AB + BC + AC = 120.

Since line segment BD bisects angle B, we can apply the angle bisector theorem.

According to the angle bisector theorem:

AD/DC = AB/BC

Given that AD = 15 and DC = 21, we can substitute these values into the equation:

15/21 = AB/BC

Now, let's solve for AB and BC.

Cross multiplying the equation:

15 * BC = 21 * AB

Dividing both sides by 21:

BC = (21 * AB) / 15

Similarly, dividing both sides by 15:

AB = (15 * BC) / 21

Now, we can substitute either AB or BC in the equation AB + BC + AC = 120 to find the other length.

Let's substitute BC in terms of AB:

AB + (21 * AB) / 15 + AC = 120

Now, we need additional information to find AC or any other relationships between the sides of the triangle.