Question

ABC and PQR are 2 identical equilateral triangles overlapping each other.
The area of the overlapping region PXBY is equal to twice the area of triangle AXP.

• A. Always
• B. Sometimes
• C. Never
• D.
• E.

Hint:

When solving problems with overlapping shapes, carefully analyze how the shapes intersect and whether the given conditions are satisfied in all possible configurations.

Answer 1

The Correct Option is B

Step 1: The problem describes two identical equilateral triangles ABC and PQR overlapping in a way that forms two distinct regions: PXBY (the overlapping region) and AXP (a smaller triangle). We need to determine if the area of PXBY is always, sometimes, or never twice the area of triangle AXP.
Step 2: Let’s analyze the geometric configuration. Since ABC and PQR are identical equilateral triangles, the relationship between their areas and the areas of the overlapping regions will depend on the exact positioning of the two triangles. If the two triangles are placed such that the overlap is exactly symmetric, the area of PXBY could be twice that of triangle AXP. However, if the triangles are shifted in other configurations, this relationship may not hold.
Step 3: Thus, the area of the overlapping region PXBY being twice the area of triangle AXP will only occur under specific conditions, making the correct answer sometimes.