Question

"One side of a triangle is 4 cm shorter than a second side. The ray bisecting the angle formed by these sides divides the opposite side into 4-cm and 6-cm segments. Find the perimeter of the triangle."

Answer 1

30 cm

Explanation:

Let the one side is x and the another side is x - 4

According to the angle bisector theorem, if a line bisects the angle of a triangle, then it divide the opposite side into the proportions of the adjacent side.

So,

`(4)/(6)=(x-4)/(x)`

4x = 6(x - 4)

4x = 6x - 24

2 x = 24

x = 12 cm

So, the sides of the triangle 12 cm, 12 - 4 = 8 cm and 4 + 6 = 10 cm

Thus, the perimeter of the triangle is 12 + 8 + 10 = 30 cm.

Answer 2

The perimeter of triangle is 30 cm.

Explanation:

Consider the provided information.

The angle bisector theorem: If the angle bisector of angle A intersect side BD at a point C. Then the ratio of the length of the line segment BC to the length of segment CD is equal to the ratio of the length of side AB to the length of side AD:

`(BC)/(CD)=(AB)/(AD)`

One side of a triangle is 4 cm shorter than a second side.

Case I: If AB is x and AD is x-4

`(4)/(6)=(x)/(x-4)`

`4x-16=6x`

`-16=2x`

Since the value of x can't be a negative number, this case is not possible.

Case II: If AB is x-4 and AD is x

`(4)/(6)=(x-4)/(x)`

`4x=6x-24`

`24=2x`

`x=12`

Hence, the length of AD is 12 cm and the length of AB is 8 cm.

The perimeter is the sum of all sides.

12+8+4+6=30 cm

Hence, the perimeter of triangle is 30 cm.