Question

line segment jl is an altitude in triangle jkm. which statement explains whether jkm is a right triangle? round measures to the nearest tenth. jkm is a right triangle because kl + lm = 15.3. jkm is a right triangle because kl + lm = 18.2. jkm is not a right triangle because kl + lm ≠ 15.3. jkm is not a right triangle because kl + lm ≠ 18.2.

Explanation:

Step1: Find the length of \(KL\) using the Pythagorean theorem in \(\triangle JKL\)

In right - triangle \(JKL\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = JK = 13\) and \(b=JL = 5\). Let \(a = KL\). Then \(KL=\sqrt{JK^{2}-JL^{2}}=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\)

Step2: Find the length of \(LM\) using the Pythagorean theorem in \(\triangle JLM\)

In right - triangle \(JLM\), by the Pythagorean theorem \(JL^{2}+LM^{2}=JM^{2}\), where \(JL = 5\) and \(JM = 8\). Then \(LM=\sqrt{JM^{2}-JL^{2}}=\sqrt{8^{2}-5^{2}}=\sqrt{64 - 25}=\sqrt{39}\approx6.2\)

Step3: Check the condition for the altitude in a right - triangle

In a right - triangle, if \(JL\) is an altitude, then \(JK^{2}+JM^{2}=KM^{2}\) (by Pythagoras) and also by the geometric mean theorem \(KL\times LM=JL^{2}\) (not relevant here). Another way: If \(\triangle JKM\) is a right - triangle with right - angle at \(J\), then \(KM=\sqrt{JK^{2}+JM^{2}}=\sqrt{13^{2}+8^{2}}=\sqrt{169 + 64}=\sqrt{233}\approx15.3\). But \(KL+LM=12 + 6.2=18.2\)

Answer:

JKM is not a right triangle because \(KL + LM
eq15.3\)