Question

modeling with mathematics you can measure the width of the lake using a surveying technique, as shown in the diagram. find the width of the lake, wx. not drawn to scale wx = m

Explanation:

Step1: Prove similarity of triangles

Since \(\angle VXW=\angle YXZ\) (vertical angles) and \(\angle VWX=\angle YZX = 90^{\circ}\), by AA (Angle - Angle) similarity criterion, \(\triangle VWX\sim\triangle YZX\).

Step2: Set up proportion

For similar triangles \(\triangle VWX\) and \(\triangle YZX\), the ratios of corresponding sides are equal. So, \(\frac{WX}{ZX}=\frac{VW}{ZY}\).
Let \(WX = x\). We know \(ZX = 6\) m, \(VW=104\) m, and \(ZY = 8\) m. Substituting these values into the proportion gives \(\frac{x}{6}=\frac{104}{8}\).

Step3: Solve for \(x\)

Cross - multiply: \(8x=104\times6\). Then \(8x = 624\). Divide both sides by 8: \(x=\frac{624}{8}=78\).

Answer:

\(78\)