Question

Two diagonals of a parallelogram intersect each other at coordinates $(17.5,\,23.5)$. Two adjacent points of the parallelogram are $(5.5,\,7.5)$ and $(13.5,\,16)$. Find the lengths of the diagonals.

• A. 15 and 30
• B. 15 and 40
• C. 17 and 30
• D. 17 and 40
• E. Multiple solutions are possible

Hint:

Use the midpoint property of parallelogram diagonals to find the opposite vertices quickly with the midpoint formula, then apply the distance formula.

Answer 1

The Correct Option is D

Diagonals of a parallelogram bisect each other. Hence $(17.5,23.5)$ is the midpoint of both diagonals.
Opposite to $A(5.5,7.5)$ is $C$ with \[ C=\big(2\cdot17.5-5.5,\;2\cdot23.5-7.5\big)=(29.5,39.5). \] Opposite to $B(13.5,16)$ is $D$ with \[ D=\big(2\cdot17.5-13.5,\;2\cdot23.5-16\big)=(21.5,31). \] Now lengths: \[ AC=\sqrt{(29.5-5.5)^2+(39.5-7.5)^2}=\sqrt{24^2+32^2}=\sqrt{1600}=40, \] \[ BD=\sqrt{(21.5-13.5)^2+(31-16)^2}=\sqrt{8^2+15^2}=\sqrt{289}=17. \] \[ \boxed{17 \text{ and } 40} \]