Question

1. to determine the distance between points x and y on the opposite sides of a river, a surveyor measures a distance of 300 m between points x and z, where z is set on the same side of the river as x. angle x-yz is measured to be 85°30′ and angle z-xy is measured to be 35°45′. compute the distance xy.

Explanation:

Step1: Convert angles to decimal degrees

First, convert \(85^\circ 30'\) and \(35^\circ 45'\) to decimal degrees.

• For \(85^\circ 30'\): \(30'=\frac{30}{60}=0.5^\circ\), so \(85^\circ 30' = 85.5^\circ\)
• For \(35^\circ 45'\): \(45'=\frac{45}{60}=0.75^\circ\), so \(35^\circ 45' = 35.75^\circ\)

Step2: Find angle at Z

In triangle \(XYZ\), the sum of angles in a triangle is \(180^\circ\). Let \(\angle XYZ = 85.5^\circ\), \(\angle ZXY = 35.75^\circ\), then \(\angle XZY=180^\circ - 85.5^\circ - 35.75^\circ=58.75^\circ\)

Step3: Apply the Law of Sines

The Law of Sines states that \(\frac{XY}{\sin\angle XZY}=\frac{XZ}{\sin\angle XYZ}\)
We know \(XZ = 300\) m, \(\angle XZY = 58.75^\circ\), \(\angle XYZ = 85.5^\circ\)
So, \(XY=\frac{XZ\times\sin\angle XZY}{\sin\angle XYZ}\)
Substitute the values: \(\sin(58.75^\circ)\approx0.855\), \(\sin(85.5^\circ)\approx0.997\)
\(XY=\frac{300\times0.855}{0.997}\approx\frac{256.5}{0.997}\approx257.37\) m

Answer:

The distance \(XY\) is approximately \(\boldsymbol{257.4}\) meters (rounded to one decimal place)