455) A triangle's bisector divides the opposite side into segments of 3 cm and 4 cm . If one of the triangle's sides is 5 cm , find the other side. 456) A triangle has a perimeter of 32 cm . A bisector divides one side into segments of 5 cm and 3 cm . Find the lengths of the triangle's sides.

We will solve these two problems using the Angle Bisector Theorem. Recall that in any triangle, an angle’s bisector divides the opposite side into segments whose lengths are proportional to the lengths of the two sides containing the angle. That is, if in triangle ABC the bisector of angle A meets side BC at D, then   AB⁄AC = BD⁄DC. ────────────────────────────── Problem 455 A triangle’s bisector divides the opposite side into segments of 3 cm and 4 cm. (That is, if BD = 3 cm and DC = 4 cm.) One of the sides forming the bisected angle is 5 cm. (That is, either AB = 5 cm or AC = 5 cm.) Since the ratio of the sides equals the ratio of the segments, we have   AB⁄AC = 3⁄4. There are two cases to consider: Case 1. Suppose the given 5 cm side is the one corresponding to the “3‐part”, that is, let   AB = 5 cm. Then since   AB = 3k  and AC = 4k, we have 3k = 5, so k = 5⁄3. Hence,   AC = 4k = 4·(5⁄3) = 20⁄3 cm. Case 2. Alternatively, if the given 5 cm side is the one corresponding to the “4‐part”, let   AC = 5 cm. Then 4k = 5, so k = 5⁄4. Thus,   AB = 3k = 3·(5⁄4) = 15⁄4 cm. Because the problem simply states “one of the triangle’s sides is 5 cm” without specifying which, the answer could be given as follows: • If the 5 cm side corresponds to the segment of length 3 (so that AB = 5 cm), then the other side is 20⁄3 cm. • If the 5 cm side corresponds to the segment of length 4 (so that AC = 5 cm), then the other side is 15⁄4 cm. Typically, by listing the segments as 3 cm and 4 cm, many texts expect the ratio to be taken in that order. Hence, one common answer is: The unknown side is (5·4⁄3) = 20⁄3 cm. ────────────────────────────── Problem 456 A triangle has a perimeter of 32 cm. A bisector divides one side into segments of 5 cm and 3 cm. (So if the bisector is drawn from angle A to side BC, then we have BD = 5 cm and DC = 3 cm, and hence BC = 5 + 3 = 8 cm.) By the Angle Bisector Theorem, the sides adjacent to angle A satisfy   AB⁄AC = BD⁄DC = 5⁄3. Write   AB = 5k   and  AC = 3k. The perimeter is   AB + AC + BC = 5k + 3k + 8 = 8k + 8. Since the perimeter is 32 cm,   8k + 8 = 32  ⇒ 8k = 24  ⇒ k = 3. Thus, the sides are:   AB = 5k = 15 cm,   AC = 3k = 9 cm, and   BC = 8 cm. ────────────────────────────── Summary of Answers 455) When an angle’s bisector divides the opposite side into 3 cm and 4 cm:  • If the 5 cm side is the one corresponding to the 3‐part, the other side is 20⁄3 cm.  • (Alternatively, if the 5 cm side corresponds to the 4‐part, the other side is 15⁄4 cm.) A common answer is: 20⁄3 cm. 456) The sides of the triangle are 15 cm, 9 cm, and 8 cm. Choose the interpretation for Problem 455 that best fits the context.