In an equilateral triangle RST, points K, L, M are located on the sides such that RK=SL=TM. Prove that triangle KLM is equilateral.
By Sarah Richards
In an equilateral triangle RST, points K, L, M are located on the sides such that RK=SL=TM.
Prove that triangle KLM is equilateral.
I have been trying to conduct this proof, but I can't seem to figure out how to show that KLM is equilateral. So far I understand that since triangle RST is equilateral, RK=SL=TM and SK=LT=MR. I have been trying to figure out how to use ASA or SAS congruence to prove the statement. Any help would be greatly appreciated.
$\endgroup$ 21 Answer
$\begingroup$You are almost there. Observe that $\triangle KSL \cong \triangle LTM \cong \triangle MRK$ through SAS rule as
- Side: $\overline{KS} = \overline{LT} = \overline{MR}$
- Angle: $\angle KSL = \angle LTM = \angle MRK = 60 ^\circ$
- Side: $\overline{SL} = \overline{TM} = \overline{RK}$
Therefore, $\overline{KL}=\overline{LM}=\overline{MK} \implies \triangle{KLM}$ is equilateral.
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