+5 As shown in the graph, the area of regular hexagon ABCDEF is 72, P is the midpoint of EF. What is the area of the shadow?
Graph: Step 1: Draw regular hexagon ABCDEF. Step 2: Connect AD. Step 3: Mark the midpoint of EF as P, and connect BP. Name the intersection between these two lines as M. Find [AFPM]
+182 Analyzing the Problem
Given:
Regular hexagon ABCDEF
Area of hexagon ABCDEF = 72
P is the midpoint of EF
We need to find the area of triangle APF (the shadow)
Approach:
Find the area of triangle ADF.
Find the ratio of the area of triangle APF to triangle ADF.
Calculate the area of triangle APF.
Solution
Step 1: Find the area of triangle ADF.
Since a regular hexagon can be divided into six equilateral triangles, the area of one equilateral triangle is 72/6 = 12.
Triangle ADF is made up of two equilateral triangles.
Area of triangle ADF = 2 * 12 = 24
Step 2: Find the ratio of the area of triangle APF to triangle ADF.
Triangle APF is similar to triangle ADF (both right triangles with the same angle at A).
The ratio of their areas is the square of the ratio of their corresponding sides.
Since P is the midpoint of EF, AP = AF/2.
The ratio of the areas of triangle APF to triangle ADF is (AF/2)^2 / AF^2 = 1/4.
Step 3: Calculate the area of triangle APF.
Area of triangle APF = (1/4) * Area of triangle ADF
Area of triangle APF = (1/4) * 24 = 6
Therefore, the area of the shadow (triangle APF) is 6 square units.
