Report

Straight Line Higher Maths The Straight Line Straight line 1 – basic examples Straight line 2 – more basic examples Straight line 3 – medians, altitudes and bisectors Straight line 4 – more on medians, altitudes and bisectors Gradient = tan of angle set(1) Gradient = tan of angle set(2) Revision Questions Basic skills and Problem Solving Questions Straight Line (1) 1. Find the gradient of the straight line passing through the points a) (2, 0) and (-1, 4) b) (-2, -3) and (3, -1) c) (6, -6) and (0, 3) d) (1, 0) and (0, -2) 2. Find the equation of the straight line passing through a) (-1, 2) with gradient 4 b) (0, 4) with gradient -2 c) (6, -2) with gradient -3 d) (5, -1) with gradient 3 3. Find the gradient of each of the following lines a) 2x + 3y - 5 = 0 b) x + y + 1 = 0 c) 4y = 3x + 1 d) 5x + 2y = 0 4. Find the equations of the lines through the following points. a) (1,3) and (3, 7) b) (2, 5) and (-1, 4) c) (0, 4) and (4, 0) d) (-1, -3) and (-2, -7) e) (6, 1) and (0, -1) f) (5, 2) and (0, -4) g) (1, 1) and (-2, -2) h) (2, 2) and (-5, 0) 5. Find the gradients of the lines perpendicular to the given lines. a) y = 4x + 3 b) y = -2x + 1 c) 2x + 3y = 1 d) 5y = x c) 6x + 3y - 7 = 0 d) 4x = 5y 6. Find the equations of the lines through a) (1, 4) and parallel to y = 4x b) (-1, -3) and parallel to 2x + 3y = 1 c) (5, 2) and perpendicular to 4y + x = 5 d) (4, 0) and perpendicular to 3x + 4y + 2 = 0 e) (-5, 4) and perpendicular to y - 2x = 0 f) (1, 1) and parallel to 2x + 5y = 1 g) (6, 1) and perpendicular to y + x + 4 = 0 Straight Line (1) Solutions 1. a) -4/3 b) 2/5 c) -3/2 d) 2 2. a) y = 4x + 6 b) y = -2x + 4 c) y = -3x + 16 d) y = 3x – 16 3. a) -2/3 b) -1 c) 3/4 d) -5/2 4. a) y = 2x + 1 c) y = -x + 4 e) x - 3y - 3 = 0 g) y = x b) x - 3y + 13 = 0 d) y = 4x + 1 f) 6x - 5y - 20 = 0 h) 2x - 7y + 10 = 0 5. a) -1/4 c) 3/2 b) 1/2 d) -5 e) 1/2 f) -5/4 6. a) y = 4x b) 2x + 3y + 11 = 0 c) y = 4x - 18 d) 4x - 3y - 16 = 0 e) x + 2y - 3 = 0 f) 2x + 5y - 7 = 0 g) y = x - 5 Straight Line (2) 1. Find the equation of the line through a) (0, -4) with gradient 5 b) (0, 2) with gradient -1 c) (3, -2) and (2, -1) d) (2, -2) and (-2, 2) 2. Find the equation of the line through a) (2, 5) parallel to y = 3x - 1 b) (-1, -2) parallel to 2x - y - 4 = 0 c) (2, 1) parallel to x + y + 1 = 0 3. Find the equation of he line passing through (-1, 1) and perpendicular to the line joining the points (4, 2) and (6, 5). 4. A triangle has vertices A(-4, 1) B(2, 2) and C(3, -2) Find a) the equations of the sides of the triangle b) the equations of the altitudes of the triangle 5. A triangle has vertices A(4, 0) B(2, -6) and C(-6, 4) Find a) the equation of AC b) the equation of the altitude through A c) the equation of the median from A to BC Straight Line (2) Solutions 1. 2. 3. 4. a) y = 5x - 4 b) y = -x + 2 c) y = -x + 1 a) y = 3x - 1 b) y = 2x c) y = -x + 3 2x + 3y - 1 = 0 a) AB x - 6y + 10 = 0 BC y = -4x + 10 AC 3x + 7y + 5 = 0 b) 5. a) b) c) x - 4y + 8 = 0 7x - 3y - 8 = 0 6x + y - 16 = 0 2x + 5y - 8 = 0 4x - 5y - 16 = 0 x - 6y - 4 = 0 d) y = -x Straight Line (3) Medians, altitudes and bisectors. 1. Given A(-3, 6) B(3,2) and C(-7, -2), find a) the equation of AB b) the equation of the altitude of ∆ABC through A c) the equation of the median of ∆ABC through B 2. Given A(8, 0) B(-2, 4) and C(-4, -2), find a) the equation of AC b) the equation of the altitude of ∆ ABC through B c) the equation of the median of ∆ ABC through C 3. Given A(5, -2) B(-2, -4) and C(-6, 2), find the equation of a) the median of ∆ABC passing through A b) the perpendicular bisector of BC c) the altitude of ∆ABC passing through B 4. The vertices of ∆ABC have coordinates (6, 2) (2, -8) and (-4, -2) respectively. Find the equation of a) the altitude through A b) the median through B c) the perpendicular bisector of AB Straight Line (3) Solutions 1. a) 2x + 3y - 12 = 0 b) 5x + 2y + 3 = 0 c) y = 2 2. a) x - 6y - 8 = 0 b) y = -6x - 8 c) 4x - 7y + 2 = 0 3. a) x + 9y + 13 = 0 b) 2x - 3y + 5 = 0 c) 11x - 4y + 6 = 0 4. a) y = x - 4 b) y = -8x + 8 c) 2x + 5y + 7 = 0 Straight Line (4) Medians, altitudes and bisectors. 1. 2. Gradient = tan In each example work out the gradient of the line and hence find the angle which the line makes with the positive direction of the x axis. y y (6,8) (5,7) 4 0 0 x y x y (0,4) 0 2 (3,0) (5,4) x 0 x y y y (4,7) (3,8) (0,4) (6,0) 0 0 x x 0 x (0,-2) y y y = 1·3x y y = -1·4x y = 0·65x+2 y = 0·7x 0 x 0 x y = 0·7x 0 x 11. Find the values of and 12. Find the size of all the angles of triangle ABC. . Find all the angles in the diagram shown. C(5,9) y y y = 0·9x y = -0.85x 0 0 x A(3,0) x B(10,0) Gradient = tan (2) In each example work out the gradient of the line and hence find the angle which the line makes with the positive direction of the x axis. (5,6) y (3,10) y y (8,3) 0 0 x x y y 0 (-3,7) (-3,4) x y (-6,2) 0 x 0 x 0 x y y (4,8) y (4,2) 0 (2,0) y 0 (1,0) x (5,4) (0,8) 0 2 x x y y (4,4) (0,5) 0 (0,-1) x 0 (7,0) x (-4,0) 0 x y y y y = 1.15x+2.5 y = 0·35x+4 0 y = 2.5x - 1 0 x x 0 x y y y y = -1.6x + 5 0 0 x 0 x y = -x x y = - 0.9x - 6 Revision Questions (Straight Line) Exam level. 1. Find the equation of the straight line which is drawn from the point of intersection of the lines 3x - y - 13 = 0 and x - 4y + 3 = 0 and which is perpendicular to 5y + 2x = 0. 2. The points (2, 6) (-4, 21) and (7, k) all lie in the same straight line. Find k. 3. A, B and C are the points (-6, -8), (11, -1) and (12, 4) respectively. a) Find the equation of AC b) If the kite ABCD is completed with AC the axis of symmetry, find the equation of BD and hence calculate the coordinates of the midpoint of BD. 4. The points A(0, -10) B(10, 3) and C(-4, 10) are the vertices of a triangle. Find a) the equation of side AC b) the equations of altitude AP and median BQ c) the coordinates of the point of intersection of AP and BQ 5. Find the equations of the straight lines through A(-2, 3) which are parallel and perpendicular to the straight line 2x - 3y + 4 = 0. If these two lines cut the line 11x + 3y = 26 at B and C respectively, find the coordinates of B and C. 6. The lines ax + 3y + 2 = 0 and 5x + 8y = 1 are perpendicular. Find the value of a. 7. The lines px + 2y - 5 = 0 and y = 2/3 x + 1 are parallel. Find the value of p. 8. Find the coordinates of the points where the line 2x - 3y = 12 cuts the x and y axes. Find the equation of the perpendicular bisector of the line joining these two points. 9. The vertices of a triangle are P(4, -1) Q(-2, -2) and R(5, 4). Find the equation of the altitude of the triangle drawn from Q. The points (3, -2) (a, 4) and (b, -5) are collinear. Show that a + 2b = 9. 11. A circle passes through the points P(2, 3) and Q(-4, -1). Find the equation of the diameter which is perpendicular to chord PQ. 12. Find the angle which the line with equation -22 x + 3y - 1 = 0 makes with the positive direction of the x axis. 13. Find the equation of the line which cuts the x axis at an angle of 135 degrees and passes through (-1, 3).