Question

An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. A second side of the triangle is 6.9 cm long. Find the longest and shortest possible lengths of the third side of the triangle. Round answers to the nearest tenth of a centimeter.
(1 point)
Responses

41.4 cm, 8.3 cm
41.4 cm, 8.3 cm

30 cm, 5.8 cm
30 cm, 5.8 cm

41.4 cm, 4.3 cm
41.4 cm, 4.3 cm

8.3 cm, 5.8 cm

Answer 1

To solve this problem, we can use the Angle Bisector Theorem. According to the theorem, if a bisector of an angle in a triangle divides the opposite side into segments of lengths a and b, then the lengths of the other two sides of the triangle are proportional to a and b, respectively.

Let the lengths of the other two sides of the triangle be x and y. According to the problem, we have:

a = 6 cm
b = 5 cm
c = 6.9 cm

Using the Angle Bisector Theorem, we can write the following proportion:

x / y = a / b

Substituting the given values, we have:

x / y = 6 / 5

Cross-multiplying, we get:

5x = 6y

Solving for y, we get:

y = (5/6)x

Now, using the triangle inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Applying this inequality to our problem, we have:

x + 6.9 > y
y + 6.9 > x

Substituting the expression for y from earlier, we get:

x + 6.9 > (5/6)x
5x + 34.5 > 6x

Simplifying this inequality, we have:

34.5 > x

x must be less than 34.5 cm.

Applying the second inequality, we get:

(5/6)x + 6.9 > x
6.9 > (1/6)x
41.4 > x

x must be greater than 41.4 cm.

Therefore, the longest possible length of the third side is 41.4 cm, and the shortest possible length is 34.5 cm.