A parallelogram ABCD has perimeter equal to 124. Let E be the foot of the perpendicular from A to BC, and let F be the foot of the perpendicular from A to CD. If AE=7 and AF=24, what is the area of the parallelogram?
To find the area of the parallelogram, let's break it down into smaller, more manageable parts.
Step 1: Determine the length of BC.
Since a parallelogram has opposite sides that are equal in length, we can conclude that BC is equal to AD. Therefore, BC = AD = 124 / 2 = 62.
Step 2: Determine the height of the parallelogram.
The height of the parallelogram can be found by considering triangle AEF. Since AE and AF are the height and base of the triangle, respectively, we can calculate the area of AEF and use it to determine the height of the parallelogram.
Area of AEF = (base * height) / 2
Area of AEF = (AF * AE) / 2
Area of AEF = (24 * 7) / 2 = 84
Since triangle AEF is right-angled, AF and AE are perpendicular to each other. Therefore, the height of the parallelogram is 84.
Step 3: Calculate the area of the parallelogram.
The area of a parallelogram can be found by multiplying its base by its height. In this case, the base is BC, and the height is 84.
Area of the parallelogram = base * height
Area of the parallelogram = BC * height
Area of the parallelogram = 62 * 84 = 5208
Therefore, the area of the parallelogram is 5208 square units.