Question

Given that LMNK is a rhombus,
KM+LN=8, and the area of KLMN is 6.

Find KL.

Answer 1

KL = √10

Explanation:

The product of KM and LN is twice the area, so is 12. This gives a quadratic in one or the other of those variables:

KM(8 -KM) = 12 . . . . the product of diagonals is twice the area

KM² -8·KM +12 = 0 . . . . rearrange to standard form

(KM -2)(KM -6) = 0 . . . . factor

KM = 2 or 6 . . . . values that make the factors zero

LN = 6 or 2 . . . . corresponding values of LN

The Pythagorean theorem tells you the side length from the diagonals:

KL² = (KM/2)² +(LN/2)² = 1² +3² = 10

KL = √10 . . . . . take the square root