Question

Let ABCD be a parallelogram, with M the midpoint of AB and N the midpoint of AD, and diagonal AC of length 36 . Let H be the intersection of AC and MN. What is the length of AH ?

Answer 1

To find the length of AH in parallelogram ABCD, we can use the properties of midpoints and similar triangles. By setting up a proportion between the lengths of AH and CD, we can solve for AH. The length of AH is 18.

Step-by-step explanation:

In a parallelogram ABCD, with M as the midpoint of AB and N as the midpoint of AD, and diagonal AC of length 36, the intersection of AC and MN is H. To find the length of AH, we can use the fact that M and N are midpoints of their respective sides. This means that MN is parallel to AB and is half the length of AB.

Since ABCD is a parallelogram, we know that opposite sides are equal in length. Therefore, AB and CD have the same length, and since MN is half the length of AB, it is also half the length of CD. So, CD is also 36.

Now, if we draw a line from H to DC, we create two similar triangles, AHM and CHD. The ratio of their corresponding sides will be the same. Since AHM and CHD share an angle at C, we can use the property of similar triangles to find the length of AH.

Let x be the length of AH. Then, using the property of similar triangles, we have:

x/36 = (x + 36)/36

Solving this equation, we find that x = 18.