A parallelogram has area P. Two triangles have areas R and T. All three figures are constructed on the same base and have the same altitude. Which relation must be true?

Introduction / Context:
This question tests a fundamental area relationship between a triangle and a parallelogram built on the same base and with the same altitude (height). The area of a parallelogram is base * height. The area of a triangle on the same base and height is (1/2) * base * height. So, for any triangle sharing the same base and altitude as the parallelogram, its area is exactly half of the parallelogram’s area. That immediately implies the parallelogram’s area is twice the triangle’s area. The variables R and T represent areas of triangles (possibly different triangles), but the key point is that each such triangle has half the parallelogram’s area under the same base and altitude condition. Therefore, P = 2R must hold (and also P = 2T, and R = T, but the question asks for one relation).

Given Data / Assumptions:

• All figures share the same base
• All figures share the same altitude (height)
• Area(parallelogram) = base * height
• Area(triangle) = (1/2) * base * height
• P is parallelogram area, R and T are triangle areas

Concept / Approach:
Compare formulas using the same base and height: P = base*height and R = (1/2)*base*height, so P = 2R.

Step-by-Step Solution:
Let the common base be b and common altitude be h Parallelogram area P = b*h Triangle area R = (1/2)*b*h So P / R = (b*h) / ((1/2)*b*h) = 2 Therefore, P = 2R

Verification / Alternative check:
If b=10 and h=6, parallelogram area is 60. Any triangle with the same base 10 and height 6 has area (1/2)*10*6 = 30. Indeed, 60 = 2*30, matching P = 2R regardless of the triangle’s slant or shape as long as base and height are the same.

Why Other Options Are Wrong:
P = R ignores the 1/2 factor in triangle area. P = R/2 reverses the correct relationship. R = 2P is the opposite of the truth. P = R + T is not guaranteed by the formulas and depends on which triangles are chosen.

Common Pitfalls:
Forgetting the 1/2 in the triangle area formula, mixing up base and altitude, or assuming shapes must be congruent rather than just sharing base and height.

Final Answer:
The relation that must be true is P = 2R.