Question

in the diagram below, $\triangle jkl$ is an equilateral triangle and $\overline{km} \perp \overline{jl}$

which statement must be true?

a. $jk = km$

b. $\triangle jkm$ is a 45 - 45 - 90 triangle.

c. $\triangle jkm$ is a 30 - 60 - 90 triangle.

d. $km = 2 \cdot jm$

Explanation:

Step1: Identify triangle properties

$\triangle JKL$ is equilateral, so $\angle J = 60^\circ$, $KM \perp JL$ means $\angle JMK = 90^\circ$.

Step2: Calculate remaining angle

In $\triangle JKM$, $\angle JKM = 180^\circ - 90^\circ - 60^\circ = 30^\circ$.

Step3: Match to triangle type

A triangle with angles $30^\circ$, $60^\circ$, $90^\circ$ is a 30-60-90 triangle.

Step4: Eliminate other options

• A: $JK$ is hypotenuse, $KM$ is shorter leg, so $JK

eq KM$.

• B: Angles are 30-60-90, not 45-45-90.
• D: In 30-60-90 triangles, hypotenuse $JK=2\cdot JM$, not $KM=2\cdot JM$.

Answer:

C. $\triangle JKM$ is a 30-60-90 triangle.