Question

in rectangle $klmn$, $km = 14sqrt{41}$ and $\frac{lm}{mn}=\frac{4}{5}$. what is the length of $overline{kl}$?

Explanation:

Step1: Let $LM = 4x$ and $MN=5x$

Set up variables based on ratio.

Step2: Apply Pythagorean theorem in right - triangle $KLM$

In rectangle $KLMN$, $KM$ is the diagonal. By Pythagorean theorem $KM^{2}=LM^{2}+MN^{2}$. Substitute $LM = 4x$ and $MN = 5x$ and $KM=14\sqrt{41}$, we get $(14\sqrt{41})^{2}=(4x)^{2}+(5x)^{2}$.

Step3: Expand and solve for $x$

Expand the equation: $196\times41 = 16x^{2}+25x^{2}=41x^{2}$. Then $x^{2}=196$, so $x = 14$ (we take the positive value since length cannot be negative).

Step4: Find the length of $KL$

In rectangle $KLMN$, $KL = MN$. Since $MN = 5x$ and $x = 14$, then $KL=70$.

Answer:

$70$