Solving Geometrical Problems Using Coordinate Geometry

4 June 2025 Using coordinate geometry LO: Use coordinate geometry to solve some geometrical problems. www.mathssupport.org

Using coordinate geometry Coordinate geometry can be used: To check the truth of a geometrical fact. To prove a geometrical fact by using general cases In these problems we find distances, midpoints, and gradients either from a sketch or by using the appropriate formulae. www.mathssupport.org

Collinear points Three or more points are collinear if they lie on the same straight line Consider the points A, B and C They all lie on the line l. C l B A = The gradient of BC= The gradient of l The gradient of AB Three points A, B and C are collinear if gradient of AB = gradient of BC www.mathssupport.org

Collinear points Example 1. Q(6, 9) and R(3, 3) Show that the points are collinear P(1,-1) 9 ( 1) 10 Gradient of PQ is = 2 = 5 6 1 3 9 3 6 Gradient of QR is 6 = 2 = 3 PQ and QR have the same gradient. Point Q is common to both segments P, Q and R are collinear. www.mathssupport.org

Using coordinate geometry Example: Q(1, 7) and R(-1, 5) Are the vertices of a triangle P(3,-1) M is the midpoint of PQ and N is the mid point of PR (a) Find the coordinates of M and N (b) Find the gradients of MN and QR (c) What can be deduced from (b)? (d) Find the distances MN and QR (e) What can be deduced from (d)? www.mathssupport.org

Using coordinate geometry Example: Q(1, 7) and R(-1, 5) Are the vertices of a triangle P(3,-1) M is the midpoint of PQ and N is the mid point of PR (a) Find the coordinates of M and N 3 + 1 2 , 1 + 7 2 M is Which is (2, 3) 3 1 2 , 1 + 5 2 N is Which is (1, 2) www.mathssupport.org

Using coordinate geometry Example: P(3,-1) Q(1, 7) and R(-1, 5) Are the vertices of a triangle M is the midpoint of PQ and N is the mid point of PR (b) Find the gradients of MN and QR M(2, 3) N(1, 2) ? ? ? ? Gradient of MN is =1 ? ? ? ? Gradient of QR is =1 www.mathssupport.org

Using coordinate geometry Example: P(3,-1) Q(1, 7) and R(-1, 5) Are the vertices of a triangle M is the midpoint of PQ and N is the mid point of PR (c) What can be deduced from (b) Gradient of MN is =1 Gradient of QR is =1 Equal gradients implies that MN is parallel to QR www.mathssupport.org

Using coordinate geometry Example: P(3,-1) Q(1, 7) and R(-1, 5) Are the vertices of a triangle M is the midpoint of PQ and N is the mid point of PR (d) Find the distances MN and QR QR = MN = ? ??+ ? ?? ? ??+ ? ?? = ? + ? = ? + ? = ? = ? = ? ? www.mathssupport.org

Using coordinate geometry Example: P(3,-1) Q(1, 7) and R(-1, 5) Are the vertices of a triangle M is the midpoint of PQ and N is the mid point of PR (e) What can be deduced from (d)? QR= ? ? MN= ? Compare and ? ? ? QR is twice as long as MN www.mathssupport.org

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