IB Maths AI SL AI 3.2 practice question

Question 1

Paper 1Hard15 marks

A council is planning a radio relay, $H$ for three flood-monitoring stations, $A$ $B$ and $C$ The relay is to be placed so that it is the same distance from each station. Figure 1 is a coordinate map; coordinates are in kilometres, with east as the positive $x$ -direction and north as the positive $y$ -direction.

The equation of the line segment $AB$ is $y=\frac13x+\frac53$ and its midpoint is $M(4,3)$

(a)

Write down the coordinates of stations $A$ and $C$ from Figure 1.

[2]

(b)

Show that the equation of the perpendicular bisector of $AB$ is $y=-3x+15$

[3]

(c)

Use your answer to part (b) and the information in Figure 1 to find the coordinates of a point that is the same distance from all three stations. Show enough working to justify that it has this property.

[7]

(d)

The relay can serve the stations only if it is no farther than $3.6\text{ km}$ from each station. Determine whether the point you found in part (c) meets this requirement. Give the distance to three significant figures.

[3]

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