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.
<span>Based in the information given in the problem, you must apply the The Angle Bisector Theorem. Let's call the triangle: "ABC"; the internal bisector of the angle that divides its opposite side: "AP"; and "x": the longest and shortest possible lengths of the third side of the triangle.
If BP= 6 cm and CP= 5 cm, we have:
BP/CP = AB/AC
We don't know if second side of the triangle (6.9 centimeters long) is AB or AC, so:
1. If AB = 6.9 cm and AC = x: 6/5 = 6.9/x x = (5x6.9)/6 x = 5.80 cm
2. If AC= 6.9 cm and AB= x: 6/5 = x/6.9 x = 6.9x6/5 x = 8.30 cm
Then, the answer is: The longest possible length of the third side of the triangle is 8.30 cm and the and shortest length of it is 5.80 cm.</span>
The use of sine and cosine could give you the angle measuments of the triangle, along with the use of tangent. Not only could it give you the angle measurements but the lengths of the sides.
