Linear coord geometry KUS objectives BAT solve linear
Linear coord geometry • KUS objectives BAT solve linear geometry problems BAT x Starter: Find the gradient of the line joining these points: A (3, 7) and B (9, 19) A (-4, 1) and B (6, -9) A (-5, 3) and B (-7, -8)
WB 16 Discussion question
WB 17 Line l 1 joins points A (3, 6) and B (6, 4) a) What is the equation of the perpendicular line through midpoint of AB ? b) Show this line goes through (3, 11/ 4) L 1 has gradient m 1 = Perpendicular line has gradient m 2 = Midpoint is = Perpendicular line: b) check: QED
WB 18 L 1 has equation 2 x + y - 6 = 0 and goes through points A(0, p) and B(q, 0) a) Find the values of p and q b) What is the equation of the perpendicular line from point C(4, 5) to line L 1 ? c) What is the area of triangle OAB? a) p = 6 and q = 3 b) Gradient L 1 = -2, gradient perpendicular = ½ c) Area = ½ base x height = ½ x 6 x 3 = 9
WB 19 Line L 1 goes through points A(-3, 2) and B(3, -1) a) Find distance AB b) Find the equation of L 1 in the form ax + by + c = 0 Perpendicular Line L 2 has equation 2 x – y + 3 = 0 and crosses L 1 at point D. c) Find coordinates of point D Line L 2 crosses the y-axis at point Q d) Find the area of triangle AQB A B
WB 19 solution II Answer Line L 1 goes through points A(-3, 2) and B(3, -1) a) b) Find distance AB Find the equation of L 1 in the form ax + by + c = 0 Perpendicular Line L 2 has equation 2 x – y + 3 = 0 and crosses L 1 at point D. c) Find coordinates of point D Line L 2 crosses the y-axis at point Q d) Find the area of triangle AQB a) AB = b) m = -½ , c) Solve simultaneous equations , D (-1, 1) d) Q(0, 3) QD = Area AQB = Q A D B
WB 20 The points A(-6, -5), B(2, -3) and C(4, -28) are the vertices of triangle ABC. Point D is the midpoint of the line joining A to B a) Show that CD is perpendicular to AB b) Find the equation of the line passing through A and B in the form ax + by + c = 0, where a, b and c are integers a) D = (-2, -4) [7] B 1 Gradient of CD = -4, Gradient of AB= ¼ M 2 Hence product of gradient is m 1 x m 2 = -1, QED A 1 b) Y + 5 = ¼ (x + 6) 2 x – 8 y – 28 = 0 or equivalent M 2 A 1
WB 21 The straight line L 1 ha equation 4 y +x = 0 The straight line L 2 has equation y = 5 x - 4 a) The lines L 1 and L 2 intersect a the point A. Calculate, as exact fractions the coordinates of A b) Find an equation of the line though A which is perpendicular to L 1. Give your answer in the form ax + by = c [6]
WB 22 The points A and B have coordinates (5, -1) and (10, 4) AB is a chord of a circle with centre C a) Find the gradient of AB The midpoint of AB is point M b) Find an equation for the line through C and M Given that the x-coordinate of point C is 6, b) Find the y coordinate of C c) Show that the radius of the circle is 17 [13]
WB 23 The points A(3, 7) B(22, 7) and C(p, q) form the vertices of a triangle. Point D(9, 2) is the midpoint of AC
WB 24 L 1 has gradient 2
Meet at (2, 1) Are perpendicular One of the gradients is 3
Summary You should be able to: • Rearrange equations of lines • Use to find the equation of a line and a perpendicular line • Solve problems involving midpoints, distances, areas, intersections and equations of lines
KUS objectives BAT solve linear geometry problems self-assess One thing learned is – One thing to improve is –
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