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SL 3.5—Intersection of lines, equations of perpendicular bisectors - IB Questionbank | RevisionDojo

Practice SL 3.5—Intersection of lines, equations of perpendicular bisectors with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 2

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 3

SLPaper 1

The equations of the sides $AB$ , $BC$ , and $CA$ of triangle $ABC$ are given as:

$AB: y=-x, \quad BC: y=2x, \quad CA: 3x+y=10$

Find the coordinates of point $C$ . Then, determine the equation of the line through $C$ that is perpendicular to line $AB$ . This line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ . Calculate the ratio $PC : CQ$ .

[7]

Question 4

SLPaper 1

The straight line passing through the points $(1, 4)$ and $(7, -2)$ intersects the line $x + 2y = 10$ at the point $P$ .

Find the coordinates of $P$ .

[5]

Question 5

SLPaper 1

Find the gradient of the line that passes through the points $\mathrm{A}(2, 3)$ and $\mathrm{B}(6, 7)$ .

[2]

Question 6

SLPaper 1

A line passes through the point $P(2, -1)$ and has a gradient of $3$ .

Find the equation of the line in the form $y = mx + c$ .

[3]

Question 7

SLPaper 2

In a city park, three security cameras are positioned at points A(1,4), B(4,1), and C(7,6) to monitor different areas. To optimize coverage, the park administration wants to find the boundary line equidistant from cameras A and B and analyze its significance for park security.

Find the midpoint of the line segment AB, which will help determine the location of equal distance between the two cameras.

[2]

Calculate the slope of the line segment AB to analyze the orientation of the cameras’ line of sight.

[2]

Determine the slope of the perpendicular bisector of AB, which would represent the line along which additional cameras could be placed for optimal coverage.

[2]

Write the equation of the perpendicular bisector in slope-intercept form ( $y=mx+b$ ) of the line segment connecting points A(1,4) and B(4,1).

[3]

Explain the geometric significance of the perpendicular bisector within the context of this park and how it relates to coverage optimization.

[3]

Question 8

SLPaper 2

Calculate the perpendicular distance from the point $P(2,5)$ to the line joining the points $A(1,-2)$ and $B(5,4)$ .

[5]

Question 9

SLPaper 1

Consider the points $A(4, 2)$ and $B(10, 8)$ .

Determine the coordinates of the midpoint of the line segment joining $A$ and $B$ .

[2]

Question 10

SLPaper 1

Let the points be $A(t, t+1)$ and $B(3t, t+3)$ .

Show that the equation of the perpendicular bisector of segment $AB$ is

y+tx=2t^2+t+2

[4]

Given that this perpendicular bisector passes through the point $(5, 2)$ , find the value of $t$ .

[2]

Paper

Level

Question 1

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 2

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 3

SLPaper 1

The equations of the sides $AB$ , $BC$ , and $CA$ of triangle $ABC$ are given as:

$AB: y=-x, \quad BC: y=2x, \quad CA: 3x+y=10$

[7]

Question 4

SLPaper 1

The straight line passing through the points $(1, 4)$ and $(7, -2)$ intersects the line $x + 2y = 10$ at the point $P$ .

Find the coordinates of $P$ .

[5]

Question 5

SLPaper 1

Find the gradient of the line that passes through the points $\mathrm{A}(2, 3)$ and $\mathrm{B}(6, 7)$ .

[2]

Question 6

SLPaper 1

A line passes through the point $P(2, -1)$ and has a gradient of $3$ .

Find the equation of the line in the form $y = mx + c$ .

[3]

Question 7

SLPaper 2

Find the midpoint of the line segment AB, which will help determine the location of equal distance between the two cameras.

[2]

Calculate the slope of the line segment AB to analyze the orientation of the cameras’ line of sight.

[2]

Determine the slope of the perpendicular bisector of AB, which would represent the line along which additional cameras could be placed for optimal coverage.

[2]

Write the equation of the perpendicular bisector in slope-intercept form ( $y=mx+b$ ) of the line segment connecting points A(1,4) and B(4,1).

[3]

Explain the geometric significance of the perpendicular bisector within the context of this park and how it relates to coverage optimization.

[3]

Question 8

SLPaper 2

Calculate the perpendicular distance from the point $P(2,5)$ to the line joining the points $A(1,-2)$ and $B(5,4)$ .

[5]

Question 9

SLPaper 1

Consider the points $A(4, 2)$ and $B(10, 8)$ .

Determine the coordinates of the midpoint of the line segment joining $A$ and $B$ .

[2]

Question 10

SLPaper 1

Let the points be $A(t, t+1)$ and $B(3t, t+3)$ .

Show that the equation of the perpendicular bisector of segment $AB$ is

y+tx=2t^2+t+2

[4]

Given that this perpendicular bisector passes through the point $(5, 2)$ , find the value of $t$ .

[2]

Paper

Level

Question 1

SLPaper 1

The line joining the points $A(2,-4)$ and $B(6,8)$ meets the $x$ - and $y$ -axes at the points $P$ and $Q$ , respectively.

Find the equation of line $AB$ and calculate the length of the segment $PQ$ .

[5]

Question 2

SLPaper 1

The line through the point $A(2,-3)$ is perpendicular to the line $2x + y = 5$ . This new line intersects the $x$ -axis at point $P$ and the $y$ -axis at point $Q$ .

Calculate the ratio $PA : AQ$ .

[7]

Question 3

SLPaper 1

The equations of the sides $AB$ , $BC$ , and $CA$ of triangle $ABC$ are given as:

$AB: y=-x, \quad BC: y=2x, \quad CA: 3x+y=10$

[7]

Question 4

SLPaper 1

The straight line passing through the points $(1, 4)$ and $(7, -2)$ intersects the line $x + 2y = 10$ at the point $P$ .

Find the coordinates of $P$ .

[5]

Question 5

SLPaper 1

Find the gradient of the line that passes through the points $\mathrm{A}(2, 3)$ and $\mathrm{B}(6, 7)$ .

[2]

Question 6

SLPaper 1

A line passes through the point $P(2, -1)$ and has a gradient of $3$ .

Find the equation of the line in the form $y = mx + c$ .

[3]

Question 7

SLPaper 2

Find the midpoint of the line segment AB, which will help determine the location of equal distance between the two cameras.

[2]

Calculate the slope of the line segment AB to analyze the orientation of the cameras’ line of sight.

[2]

Determine the slope of the perpendicular bisector of AB, which would represent the line along which additional cameras could be placed for optimal coverage.

[2]

Write the equation of the perpendicular bisector in slope-intercept form ( $y=mx+b$ ) of the line segment connecting points A(1,4) and B(4,1).

[3]

Explain the geometric significance of the perpendicular bisector within the context of this park and how it relates to coverage optimization.

[3]

Question 8

SLPaper 2

Calculate the perpendicular distance from the point $P(2,5)$ to the line joining the points $A(1,-2)$ and $B(5,4)$ .

[5]

Question 9

SLPaper 1

Consider the points $A(4, 2)$ and $B(10, 8)$ .

Determine the coordinates of the midpoint of the line segment joining $A$ and $B$ .

[2]

Question 10

SLPaper 1

Let the points be $A(t, t+1)$ and $B(3t, t+3)$ .

Show that the equation of the perpendicular bisector of segment $AB$ is

y+tx=2t^2+t+2

[4]

Given that this perpendicular bisector passes through the point $(5, 2)$ , find the value of $t$ .

[2]

SL 3.5—Intersection of lines, equations of perpendicular bisectors Questionbank