Question

A triangle and a parallelogram are on the same base and between the same parallel lines they have:

• A. Equal area
• B. Equal perimeter
• C. Both
• D. None of these

Answer 1

Answer: (D)

None of these

$\Rightarrow$ Let $△$ABP and a parallelogram ABCD be on the base AB and between the same parallels AB abd PC.

$\Rightarrow$ Draw BQ$\parallel$AP to obtain another parallelogram ABQP and ABCD are on the same base AB and between the same parallel AB and PC.

$\therefore$ $ar\left(ABQP\right)=ar\left(ABCD\right)$ ---------- ( 1 )

$\Rightarrow$ But $△$PAB $\cong$ $△$BQP [Diagonals PB divides parallelogram ABQP into two congruent triangles.]

$\Rightarrow$ So $ar\left(PAB\right)=ar\left(BQP\right)$ ----------- ( 2 )

$\therefore$ $ar(PAB)=ar(ABQP) ----------- ( 3 ) [From ( 2 )]

This gives $ar\left(PAB\right)=\frac{1}{2}ar\left(ABCD\right)$ [From ( 1 ) and ( 3 )]

$\Rightarrow$ So, we have prove that a triangle and a parallelogram are on same base and between the same parallel lines they don't have equal area or equal perimeter.

$\Rightarrow$ So, correct answer is option D.