Question

question
find the measure of $overline{lm}$.
answer
attempt 1 out of 2
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Explanation:

Step1: Identify applicable law

This is a triangle with two sides and a pair of congruent angles (marked at K and M), so we use the Law of Sines: $\frac{LM}{\sin K} = \frac{KM}{\sin L} = \frac{KL}{\sin M}$. Since $\angle K = \angle M$, let $\angle K = \angle M = \theta$. First find $\angle L$.

Step2: Calculate $\angle L$

Sum of angles in a triangle is $180^\circ$.
$\angle L = 180^\circ - 2\theta$, but we can directly apply Law of Sines to the known sides:
$\frac{LM}{KM} = \frac{KL}{LM}$ (because $\sin K = \sin M$)

Step3: Rearrange to solve for $LM$

Cross-multiply the proportion:
$LM^2 = KL \times KM$
Substitute $KL=25$, $KM=40$:
$LM^2 = 25 \times 40 = 1000$

Step4: Compute square root

$LM = \sqrt{1000} = \sqrt{100 \times 10} = 10\sqrt{10}$

Answer:

$10\sqrt{10}$ (or approximately 31.62 if a decimal is needed)