Question

to indirectly measure the distance across a lake, ethan makes use of a couple landmarks at points ( l ) and ( m ). he measures ( kn ), ( nl ), and ( no ) as marked. find the distance across the lake (( lm )), rounding your answer to the nearest hundredth of a meter. (diagram is not to scale.)

Explanation:

Step1: Prove similarity of triangles

Since \(\angle KLN=\angle KNO = 90^{\circ}\) and \(\angle K\) is common to \(\triangle KLM\) and \(\triangle KNO\), by the AA (Angle - Angle) similarity criterion, \(\triangle KLM\sim\triangle KNO\).

Step2: Set up proportion

For similar triangles \(\triangle KLM\) and \(\triangle KNO\), the ratios of corresponding sides are equal. That is \(\frac{LM}{NO}=\frac{KL}{KN}\).
We know that \(KL=KN + NL=95 + 95=190\) m, \(KN = 95\) m, and \(NO = 89.9\) m.
Substituting the values into the proportion \(\frac{LM}{89.9}=\frac{190}{95}\).

Step3: Solve for \(LM\)

Cross - multiply: \(95\times LM=190\times89.9\).
Then \(LM=\frac{190\times89.9}{95}\).
Since \(\frac{190}{95}=2\), so \(LM = 2\times89.9=179.8\) m.

Answer:

\(179.80\) m