Question

1. a sports car is 14 km south of an intersection. a van is 14 km west of the same intersection. a) draw a diagram to illustrate the problem. use trigonometry to find an exact expression for the distance between the two vehicles.

Explanation:

Step1: Identify the right - angled triangle

The positions of the sports car, van, and intersection form a right - angled triangle. Let the distance of the sports car from the intersection be $a = 14$ km (south - direction) and the distance of the van from the intersection be $b = 14$ km (west - direction). The distance between the two vehicles is the hypotenuse $c$ of the right - angled triangle.

Step2: Apply the Pythagorean theorem

In a right - angled triangle, $c^{2}=a^{2}+b^{2}$. Substituting $a = 14$ and $b = 14$ into the formula, we get $c^{2}=14^{2}+14^{2}=196 + 196=2\times196$.

Step3: Solve for $c$

Taking the square root of both sides, $c=\sqrt{2\times196}=\sqrt{2}\times\sqrt{196}=14\sqrt{2}$ km.

Answer:

The distance between the two vehicles is $14\sqrt{2}$ km.