Question

Please help with this math problem!!! 25 points

What is value of x? Enter your answer in the box. x =
A triangle with vertices labeled as A, B, and C. Side B C is the base. A line segment is drawn from vertex A to base B C that bisects angle B A C into two parts marked with single arcs. Side A B is labeled as 8. Side A C is labeled as 12. Side B C is divided into two equal parts labeled as x plus 4 and 2 x plus 1.

here is the picture of the problem

Answer 1

By Triangle-Angle-Bisector Theorem,
8:12=(x+4):(2x+1)

(x+4):(2x+1)=2:3
3(x+4)=2(2x+1)
3x+12=4x+2
x=10

Answer 2

10

Explanation:

The triangle angle bisector theorem states that an angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

This means the proportion of the two given sides of ABC, 8/12, is the same as the proportion of the segments that make up BC, (x+4)/(2x+1). This gives us the proportion

8/12 = (x+4)/(2x+1)

Cross multiplying,

8(2x+1) = 12(x+4)

Using the distributive property,

8(2x)+8(1) = 12(x)+12(4)

16x+8 = 12x+48

Subtract 12x from each side:

16x+8-12x = 12x+48-12x

4x+8 = 48

Subtract 8 from each side:

4x+8-8 = 48-8

4x = 40

Divide both sides by 4:

4x/4 = 40/4

x = 10