Question

in the rectangle below, $jn = 3x + 5$, $ln=6x - 7$, and $mangle jnm = 90^{circ}$. find $km$ and $mangle nlk$.

Explanation:

Step1: Use property of rectangle diagonals

In a rectangle, the diagonals are equal and bisect each other. So $JN = LN$.
Set up the equation $3x + 5=6x - 7$.

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $5 = 3x-7$.
Add 7 to both sides: $3x=12$, then $x = 4$.

Step3: Find the length of $JN$

Substitute $x = 4$ into the expression for $JN$: $JN=3\times4 + 5=17$.
Since $KM = 2JN$ (diagonals bisect each other), $KM=2\times17 = 34$.

Step4: Find the measure of $\angle NLK$

In a rectangle, each angle is $90^{\circ}$. Diagonals of a rectangle are equal and bisect each other. $\angle JNM = 98^{\circ}$, but this is likely a mis - label as in a rectangle, the diagonals form congruent isosceles triangles at the intersection. In a rectangle, $\angle NLK$ and $\angle NKL$ are congruent. And $\angle KNL= 180^{\circ}-98^{\circ}=82^{\circ}$. Since $\triangle KNL$ is isosceles ($KN = NL$), $\angle NLK=\frac{180^{\circ}-82^{\circ}}{2}=49^{\circ}$.

Answer:

$KM = 34$, $m\angle NLK=49^{\circ}$