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Straight Line · Guided Practice

Straight Line

18 questions with answers and video solutions. Try each one before revealing the answer.

1
Find the equation of the straight line through (4,−1)\displaystyle (4,-1) that is parallel to the line with equation 2x+y=5.\displaystyle 2x+y=5.
2
A and B are the points (−8,7)\displaystyle (-8,7) and (2,t).\displaystyle (2,t).
The line AB is parallel to the line with equation 5y−x=9.\displaystyle 5y-x=9.
Determine the value of t.\displaystyle t.
3
Line l1\displaystyle l_1 has equation 3x−4y=1.\displaystyle 3x-4y=1.
Line l2\displaystyle l_2 is perpendicular to l1.\displaystyle l_1.
The two lines intersect at (3,2).\displaystyle (3,2).
Determine the equation of l2.\displaystyle l_2.
4
Three points A, B and C are defined as (1,−5)\displaystyle (1,-5), (10,7)\displaystyle (10,7) and (4,−1)\displaystyle (4,-1) respectively.
Are A, B and C collinear?
Justify your answer.
5
Three points P, Q and R are defined as (−1,2)\displaystyle (-1,2), (2,8)\displaystyle (2,8) and (−4,−7)\displaystyle (-4,-7) respectively.
Are P, Q and R collinear?
Justify your answer.
6
The line l1\displaystyle l_1 makes an angle of 30∘\displaystyle 30^\circ with the positive direction of the x\displaystyle x-axis.
Find the equation of the line l2\displaystyle l_2 which is perpendicular to l1\displaystyle l_1 and passes through the point (−3,23).\displaystyle (-3,2\sqrt{3}).
7
The line l1\displaystyle l_1 has a negative gradient and makes an angle of 30∘\displaystyle 30^\circ with the negative direction of the x\displaystyle x-axis.
Find the equation of the line l2\displaystyle l_2 which is perpendicular to l1\displaystyle l_1 and passes through the point (−3,23).\displaystyle (-3,2\sqrt{3}).
8
Determine the acute angle that the line with equation 2x−3y=1\displaystyle 2x-3y=1 makes with the y\displaystyle y-axis.
9
P, Q and R are the points (2,−1)\displaystyle (2,-1), (6,7)\displaystyle (6,7) and (3,−2)\displaystyle (3,-2) respectively.
In triangle PQR, determine the equation of the median through R.
10
A is (−3,4)\displaystyle (-3,4) and B is (7,−6).\displaystyle (7,-6).
Determine the equation of the perpendicular bisector of AB.
11
In triangle KLM, the vertices K, L and M are the points (−4,1)\displaystyle (-4,1), (1,7)\displaystyle (1,7) and (3,3)\displaystyle (3,3) respectively.
(a)  Find the equation of the altitude from K.
(b)  Determine the coordinates of the point where the altitude from K intersects the straight line through L and M.
12
2017 P1 Q7
3 Marks
A(−3,5)\displaystyle A(-3,5), B(7,9)\displaystyle B(7, 9) and C(2,11)\displaystyle C(2,11) are the vertices of a triangle.
Find the equation of the median through C.
13
2021 P1 Q4
Determine whether the line passing through (−4,2)\displaystyle (-4,2) and (2,−7)\displaystyle (2,-7) is perpendicular to the line with equation 3y=2x+9.\displaystyle 3y=2x+9.
14
2023 P2 Q1
(3, 2)5 Marks

Triangle PQR has vertices P(5,−1)\displaystyle P(5,-1), Q(−2,8)\displaystyle Q(-2,8) and R(13, 3).

Triangle PQR

(a) Find the equation of the altitude from P.
(b) Calculate the angle that the side PR makes with the positive direction of the x-axis.

15
2024 P2 Q1
(3, 3, 2)8 Marks

Triangle ABC has vertices A(−3,8)\displaystyle A(-3,8), B(−1,−6)\displaystyle B(-1,-6) and C(11,0)\displaystyle C(11,0).

Triangle ABC with median and perpendicular line

(a)  Find the equation of the median through B.
(b)  Find the equation of L, the line perpendicular to BC passing through C.
(c)  Determine the coordinates of the point of intersection of the median through B and the line L.

16
2025 P1 Q2
4 Marks
Find the equation of the perpendicular bisector of the line joining A(1,4)\displaystyle A(1,4) and B(9,10).\displaystyle B(9,10).
17
2025 P2 Q1
(3, 3, 2)8 Marks

Triangle ABC has vertices A(−9,−14),\displaystyle A(-9,-14), B(9,20)\displaystyle B(9,20) and C(21,−24).\displaystyle C(21,-24).

Triangle ABC

(a) Find the equation of the altitude through B.
(b) Find the equation of the median through A.
(c) Determine the point of intersection of the altitude through B and the median through A.

18
2026 P2 Q8
(4, 2)6 Marks

A and B are the points (−2,11)\displaystyle (-2,11) and (10,−13).\displaystyle (10,-13).

Points A, B and C with the line L, the perpendicular bisector of AB, and the line AC

(a)  Find the equation of L, the perpendicular bisector of AB.
The equation of AC is y+x=9.\displaystyle y+x=9.
(b)  Calculate the size of the obtuse angle between L and AC.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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