Question

Points K and L lie on side BC, while point M lies on side AD of rectangle ABCD. It is known that segment KM is a bisector of ∠AKC and segment LM is a bisector of ∠BLD. Given AK=6, KL=7, and LD=8. Find the lengths of segments BK and LC.

Answer 1

To find the lengths of segments BK and LC, we can set up proportions using the angles of the bisectors. Then, we substitute the given values and solve for the unknown lengths.

Step-by-step explanation:

Let's start by setting up a proportion using the angles of the bisectors: ∠AKC and ∠BLD. Since KM is a bisector of ∠AKC, we can set up the proportion: AK/KC = AM/MC. And since LM is a bisector of ∠BLD, we can set up the proportion: LD/DC = LM/MB.

Given that AK = 6, KL = 7, and LD = 8, we can substitute these values into the proportions: 6/KC = AM/MC and 8/DC = LM/MB.

Since KL = 7 and KC = MC - MK, we can substitute these values and solve for MC: 7 = MC - 3. Plugging the value of MC back into the proportion, we can solve for AM: 6/(MC - 3) = AM/MC.

Now, using the value of 8/DC = LM/MB, and knowing that DC = 15, we can solve for MB: 8/15 = LM/MB.

Using the values of AM and MB, we can solve for AK and KL: AK = AM + MK and KL = KM + ML.