Question

In rectangle ABCD, the angle bisector of ∠A intersects side DC at point M and the angle bisector of ∠C intersects side AB at point N. The length of CM is equal to the length of AM and it is 6 in longer than the length of DM. Find the perimeter of quadrilateral ANCM.

Answer 1

In rectangle ABCD, if CM is equal to AM and 6 inches longer than DM, we can find the lengths of CM, AM, and DM. The perimeter of quadrilateral ANCM can be found by adding the lengths of all four sides: AB + BC + CM + AN. Using the given information, we can calculate that the perimeter of quadrilateral ANCM is 42 inches.

Step-by-step explanation:

In rectangle ABCD, let CM = x. Since CM is equal to AM and 6 inches longer than DM, we can determine the lengths of CM, AM, and DM.

Since the angle bisector of ∠A intersects side DC at point M, we can use the angle bisector theorem to find the ratios of the lengths of CM and DM to the lengths of AB and BC respectively:

CM/DM = AB/BC

Since length CM is equal to length AM, we have:

CM/AM = AB/BC

Let's assume that CM = x. From the above equation, we can write:

x/(x - 6) = AB/BC

Since ABCD is a rectangle, AB = BC. Therefore, AB/BC = 1 and the equation becomes:

x/(x - 6) = 1

By solving this equation, we find that x = 12. Therefore, CM = AM = 12 and DM = 6.

The perimeter of quadrilateral ANCM is the sum of the lengths of all four sides: AB + BC + CM + AN.

Since AB = BC = 12 and CM = AM = 12, and N is the intersection of the angle bisector of ∠C and side AB, we can conclude that AN = NC = AB/2 = 6.

Therefore, the perimeter of quadrilateral ANCM is 12 + 12 + 12 + 6 = 42 inches.

Answer 2

Perimeter of quadrilateral ANCM = AN+NC+CM+MA

CM=AM = x in
Therefore, AN=NC = x in
In other words, all sides of the new quadrilateral are equal.

Perimeter (P) = 4x in

Now, Cos 45 = (x-6)/x =>0.7041 = (x-6)/x => 0.7041x=x-6 =>x= 6/(1-0.70741)
x= 20.5 in

Therefore, P=4*20.5 = 82 in