The points X, Y and Z are located such that Y is 15 km south of X, Z is 20 km from X on a bearing of 270".
(a)
To find |YZ|, we need to first draw a diagram of the points X, Y, and Z. Since Y is 15 km south of X, we can draw a line segment from X going south with a length of 15 km, and label the endpoint as Y. Next, we know that Z is 20 km from X on a bearing of 270°, which means Z is directly west of X. Therefore, we can draw a line segment from X going west with a length of 20 km, and label the endpoint as Z. Finally, we can draw a line segment from Y to Z.
To find the length of |YZ|, we can use the Pythagorean theorem, which states that for a right triangle with legs of lengths a and b and hypotenuse of length c, \(c^2 = a^2 + b^2\). In this case, YZ is the hypotenuse, and its length can be found by:
\(|YZ|^2 = |XY|^2 + |XZ|^2\)
where \(|XY|\) is the length of the segment from X to Y, and \(|XZ|\) is the length of the segment from X to Z.
We know that \(|XY|\) is 15 km, and \(|XZ|\) is 20 km, so:
\(|YZ|^2 = 15^2 + 20^2\) \(|YZ|^2 = 225 + 400\) \(|YZ|^2 = 625\) \(|YZ| = \sqrt{625}\) \(|YZ| = 25\) km (rounded to two significant figures)
Therefore, the length of \(|YZ|\) is 25 km, correct to two significant figures.
(b)
To find the bearing of Y from Z, we can use trigonometry. Specifically, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, we can use the tangent of the angle formed by the line segment YZ and the horizontal axis to find the bearing of Y from Z.
Let \(\theta\) be the angle formed by the line segment YZ and the horizontal axis. Then:
\(\tan(\theta) = \frac{{\text{{opposite}}}}{{\text{{adjacent}}}} = \frac{{|XY|}}{{|XZ|}}\)
We know that \(|XY|\) is 15 km and \(|XZ|\) is 20 km, so:
\(\tan(\theta) = \frac{{15}}{{20}}\) \(\tan(\theta) = 0.75\)
To find \(\theta\), we can take the inverse tangent (also called arctangent) of both sides:
\(\theta = \tan^{-1}(0.75)\) \(\theta = 36.87^\circ\)
Therefore, the bearing of Y from Z is 37°, rounded to the nearest degree. This means that if you were standing at point Z and facing due north, you would need to turn