Question

in the rectangle below, $km = 4x + 2$, $jl = 5x - 5$, and $m\angle nmj = 48^\circ$. find $jn$ and $m\angle knl$.

Explanation:

Step1: Set diagonals equal (rectangle property)

In a rectangle, diagonals are congruent, so $KM = JL$.
$4x + 2 = 5x - 5$

Step2: Solve for $x$

Rearrange to isolate $x$:
$2 + 5 = 5x - 4x$
$x = 7$

Step3: Calculate length of diagonal $JL$

Substitute $x=7$ into $JL = 5x - 5$:
$JL = 5(7) - 5 = 35 - 5 = 30$

Step4: Find $JN$ (half the diagonal)

Rectangle diagonals bisect each other, so $JN = \frac{1}{2}JL$:
$JN = \frac{1}{2} \times 30 = 15$

Step5: Analyze $\triangle NMJ$ (isosceles triangle)

$JN = NM$, so $\angle NMJ = \angle NJM = 48^\circ$

Step6: Find $\angle JNM$ (triangle angle sum)

$\angle JNM = 180^\circ - 48^\circ - 48^\circ = 84^\circ$

Step7: Find $m\angle KNL$ (vertical angle)

$\angle KNL$ and $\angle JNM$ are vertical angles, so they are congruent:
$m\angle KNL = 84^\circ$

Answer:

$JN = 15$
$m\angle KNL = 84^\circ$