In a parallelogram $ ABCD $, the length of the line joining the midpoints of $ AD $ and $ AC $ is $ 2 $ units.

Question

In a parallelogram $ ABCD $, the length of the line joining the midpoints of $ AD $ and $ AC $ is $ 2 $ units. If the perimeter of the parallelogram is $ 26 $ units, then find the length of $ AD $.

Examoogle · Accepted Answer

The correct answer is Option D: 9 units. In triangle ADC, the line joining midpoints of AD and AC is parallel to DC and half its length. Thus, DC = 2 × 2 = 4. Since perimeter = 2(AD + DC) = 26, then AD + 4 = 13, so AD = 9.

Step-by-Step Solution & Concepts:
Shortcut Trick: In triangle ADC, the line joining midpoints of AD and AC is parallel to DC and half its length. Thus, DC = 2 × 2 = 4. Since perimeter = 2(AD + DC) = 26, then AD + 4 = 13, so AD = 9.

Formula Used: $$P = 2(l + w)$$, $$midsegment = \frac{1}{2} 	imes side$$

Given: Parallelogram ABCD, perimeter = 26 units, length of line joining midpoints of AD and AC = 2 units.

Steps: Analyze Triangle ADC: In $	riangle ADC$, let $M$ be the midpoint of $AD$ and $N$ be the midpoint of $AC$. The line segment $MN$ connects these midpoints.. Determine base CD: By the Midpoint Theorem, the segment connecting the midpoints of two sides of a triangle is half the length of the third side (DC).. Calculate perimeter: The perimeter of parallelogram $ABCD$ is $2(AD + DC) = 26$. Substituting $DC = 4$:. Solve for AD: Subtract 4 from 13 to find the side length $AD$.

Answer: D is correct because by the Midpoint Theorem, $DC = 2 	imes 2 = 4$ and given the perimeter $2(AD + DC) = 26$, we get $AD + 4 = 13$, so $AD = 9$.

Exam Tip: The Midpoint Theorem is a fundamental bridge between triangle properties and quadrilateral properties; always look for triangles within complex shapes in competitive exams.

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