Answer:
4y-2x+6
Explanation:
Let's denote the length of CM and AM as x. Since CM is 6 inches longer than DM, we can express DM's length as x - 6.
Since the angle bisector of angle A intersects side DC at point M, and the angle bisector of angle C intersects side AB at point N, we can conclude that triangles ANM and CDM are similar triangles.
According to the similarity of triangles, we can write the following proportion:
AN / CD = AM / DM
Substituting the given values, we have:
AN / CD = x / (x - 6)
Now, we know that rectangle ABCD has equal opposite sides. Therefore, CD is equal to AB. Let's denote their common length as y.
We can express the lengths AN and CD in terms of y using the similarity proportion:
AN = (x / (x - 6)) * CD
AN = (x / (x - 6)) * y
Since the opposite sides of a rectangle are equal, AB = CD = y.
Now, let's find the perimeter of quadrilateral ANCM:
Perimeter = AB + BC + CD + DA
Perimeter = y + BC + y + DA
Perimeter = 2y + BC + DA
To find BC and DA, we need to consider the fact that the angle bisector of angle A intersects side DC at point M. Since CM is equal to AM, we can divide side DC into two equal parts, each with length x. Therefore, DM has a length of (x - 6).
Similarly, since the angle bisector of angle C intersects side AB at point N, we can divide side AB into two equal parts, each with length x. Therefore, AN has a length of x.
BC = AB - AM = y - x
DA = DC - DM = y - (x - 6) = y - x + 6
Substituting the values of BC and DA into the perimeter equation, we have:
Perimeter = 2y + (y - x) + (y - x + 6)
Perimeter = 4y - 2x + 6
Therefore, the perimeter of quadrilateral ANCM is 4y - 2x + 6.