IB Maths AI HL Sl 3 5 Perpendicular Bisectors Questions

Practise finding midpoints and gradients, constructing perpendicular bisectors, and checking equal-distance or perpendicularity conditions in the resulting region.

Exam points

Find the midpoint and gradient of a segment and construct the perpendicular bisector equation.
Use two points or a segment equation plus midpoint to determine a perpendicular-bisector boundary.
Check the line by equal-distance or perpendicularity reasoning and interpret the resulting region.

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Study SL 3.5—Perpendicular bisectors conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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This question considers how the assessment of the Air Quality Index (AQI) for a school depends on the method chosen by the person doing the assessing.
Air quality for a district is measured at three monitoring stations. The positions of these stations on a coordinate system with units in kilometres are A(0,5), B(8,9) and C(8,1).
A Voronoi diagram is constructed with the three stations as sites.

Figure for Question IB Maths AI HL Sl 3 5 Perpendicular Bisectors Questions question 1 — IB Maths AI HL

(a)

Find the equation of the perpendicular bisector of [BC].

[ 2 ]

(b)

Find the mid-point of [SB].

[ 2 ]

attempt to find a midpoint

\left(\frac{5+8}{2}, \frac{6+9}{2}\right)

(6.5,7.5)

(c)

Show that the equation of the perpendicular bisector of $[\mathrm{SB}]$ is y=14-x.

[ 3 ]

gradient is $\frac{9-6}{8-5}=1$
gradient of perpendicular bisector is -1
substituting their gradient and their mid-point into the equation of a line

y-7.5=-1(x-6.5) \text { OR } 7.5=-1 \times 6.5+c

y=14-x

Note: If A1A1M1 have been awarded but there is then incorrect work after the final M1 such that the AG does not follow, do not award the final M1.

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IB Maths AI HL Sl 3 5 Perpendicular Bisectors Questions question 1

Question (a)

Question (b)

Question (c)