IBDP Maths AI: SL 3.5 Equations of perpendicular bisectors Exam Style Questions Paper 1

Question

Points A (3,4), B (9,6), and C (11,2) are shown on the following diagram, along with the perpendicular bisectors of [AB], [AC], and [BC].

The perpendicular bisector of [BC] intercepts the axes at coordinates (0,−1) and (2,0).

(a) Write down the equation of the perpendicular bisector of [BC].

The equation of the perpendicular bisector of [AB] is \( y = -3x + 23 \).

(b) Find the coordinates of point V where the perpendicular bisectors meet. Give your answer to four significant figures.

A Voronoi diagram is constructed with points A, B, and C as the three sites.

(c) Draw, clearly, the edges of the Voronoi diagram on the given diagram.

▶️Answer/Explanation

Detailed Solution

(a) Finding the Equation of the Perpendicular Bisector of [BC]

The perpendicular bisector passes through the x-axis at (2,0) and the y-axis at (0,-1). Using the two given points, we find the slope:

\[ \text{Slope} = \frac{0 – (-1)}{2 – 0} = \frac{1}{2} \]

The equation of the line in slope-intercept form \( y = mx + c \) is:

\[ y = 0.5x – 1 \]

(b) Finding the Intersection Point of the Perpendicular Bisectors

We solve for \( V \) where the perpendicular bisectors of [AB] and [BC] intersect.

Given equations:

Setting them equal:

\[ -3x + 23 = 0.5x – 1 \]

Solving for \( x \):

\[ 23 + 1 = 0.5x + 3x \]

\[ 24 = 3.5x \]

\[ x = \frac{24}{3.5} = 6.857 \]

Substituting \( x = 6.857 \) into \( y = 0.5x – 1 \):

\[ y = 0.5(6.857) – 1 \]

\[ y = 3.429 – 1 = 2.429 \]

So, the coordinates of \( V \) are (6.857, 2.429).

(c) Voronoi Diagram

The Voronoi diagram is drawn by extending the perpendicular bisectors to form regions. The updated diagram is shown below:

……………………………Markscheme……………………………….

(a)

\( y = 0.5x – 1 \)

(b)

\( (6.857, 2.429) \)

(c)

Correctly drawn Voronoi diagram with edges clearly marked.

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