Question

in the diagram below, $\triangle jkl$ is an equilateral triangle and $overline{km}perpoverline{jl}$. which statement must be true? a. $\triangle jkm$ is a 45 - 45 - 90 triangle. b. $\triangle jkm$ is a 30 - 60 - 90 triangle. c. $km = 2cdot jm$ d. $jk = km$

Explanation:

Step1: Recall properties of equilateral triangle

In equilateral triangle $\triangle{JKL}$, each angle is $60^{\circ}$, so $\angle{J}=60^{\circ}$. Since $KM\perp JL$, $\angle{KMJ} = 90^{\circ}$.

Step2: Calculate third - angle of $\triangle{JKM}$

In $\triangle{JKM}$, using the angle - sum property of a triangle ($\angle{J}+\angle{KMJ}+\angle{JKM}=180^{\circ}$), we substitute $\angle{J} = 60^{\circ}$ and $\angle{KMJ}=90^{\circ}$. Then $60^{\circ}+90^{\circ}+\angle{JKM}=180^{\circ}$, so $\angle{JKM}=30^{\circ}$. Thus, $\triangle{JKM}$ is a 30 - 60 - 90 triangle.

Step3: Analyze side - length relationships in 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle $\triangle{JKM}$, the side opposite the 30 - degree angle is $JM$, the side opposite the 60 - degree angle is $KM$, and the hypotenuse is $JK$. The relationship between the sides is $JK = 2JM$ and $KM=\sqrt{3}JM$.

Answer:

B. $\triangle{JKM}$ is a 30 - 60 - 90 triangle.