If a parallelogram with area $P$, a rectangle with area $R$ and a triangle with area $T$ are all constructed on the same base and all have the same altitude, then which of the following statements is false?

### Concept & Logic Compare the fundamental area formulas for shapes constructed on a common base ($b$) with a common altitude ($h$). $$P = b \times h$$ $$R = b \times h$$ $$T = \frac{1}{2} \times b \times h$$ ### Step-by-Step Solution 1. **Establish relationships:** From the formulas, it is clear that $P = R$. Also, since $T = 0.5 \times (b \times h)$, it follows that $T = 0.5 \times P$ (or $P = 2T$) and $T = 0.5 \times R$. 2. **Evaluate Option (a):** $P = R$. This is **True**. 3. **Evaluate Option (c):** $P = 2T$. This is **True**. 4. **Evaluate Option (d):** $T = \frac{1}{2} R$. This is **True**. 5. **Evaluate Option (b):** $P + T = 2R$. Substitute $P$ and $T$ in terms of $R$: $R + 0.5R = 1.5R$. The statement claims it equals $2R$. Since $1.5R \neq 2R$, this statement is **False**. ### Exam Strategy & Shortcut Assign simple numerical values! Let base = 10 and height = 10. Then $P = 100$, $R = 100$, and $T = 50$. Check (b): $100 + 50 = 150$. Does $150 = 2(100)$? No. The false statement is instantly identified. ### Common Pitfall Misreading the question and selecting a *true* statement as the answer instead of correctly identifying the *false* one. ### Final Answer Therefore, the correct answer is **$P + T = 2R$**.