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.
41.4 cm, 8.3 cm
30 cm, 5.8 cm
41.4 cm, 4.3 cm
8.3 cm, 5.8 cm
i believe the answer to be D.
call the third side of the triangle x.
The angle bisector theorem states that the angle bisector to a side divides the side into segments whose ratio equals the ratio of the other two sides.
Since we don't know which other side is 6.9, either
x/6.9 = 5/6
or
x/6.9 = 6/5
So, x is 5.8 or 8.3 -- answer D
in awe at the size of this lad, absolute unit
To solve this problem, we can use the Angle Bisector Theorem, which states that the angle bisector of a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.
Let's represent the sides of the triangle as follows:
AB = 6.9 cm (the second side of the triangle)
AC = 6 cm (segment of the opposite side given)
BC = 5 cm (segment of the opposite side given)
Let the third side of the triangle be x cm.
Now, let's apply the Angle Bisector Theorem:
AC/BC = AB/BC
6/5 = 6.9/x
Cross-multiplying, we get:
6x = 5 * 6.9
6x = 34.5
x = 34.5/6
x ≈ 5.8 cm
Therefore, the shortest possible length of the third side of the triangle is approximately 5.8 cm.
To find the longest possible length of the third side, we can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, the sum of the lengths of the two shorter sides (5 cm and 6 cm) must be greater than the length of the third side (x cm):
5 + 6 > x
11 > x
Therefore, the longest possible length of the third side of the triangle is 11 cm.
Rounding to the nearest tenth of a centimeter, we have:
Shortest possible length: approximately 5.8 cm
Longest possible length: approximately 11 cm
Therefore, the correct answer is:
8.3 cm (rounded from 5.8 cm) and 41.4 cm (rounded from 11 cm).
So, the answer is option D: 8.3 cm, 5.8 cm.