1. SCHOLAR Higher Mathematics Homework Session Thursday 22 nd October 6 - 7pm You will need a pencil, paper and a calculator for some of the activities

2. My name is Margaret Ferguson I am the SCHOLAR online tutor for Maths

3. The Straight Line Tonight’s Session will cover

4. Given any two points (x 1,y 1 ) and (x 2,y 2 ) The distance formula is D = The gradient formula is m = The midpoint formula is A horizontal line has gradient 0 The equation of a horizontal straight line which passes through the point (a,b) is y = b A vertical straight line has an undefined gradient The equation of a vertical straight line which passes through the point (a,b) is x = a The equation of a straight line with gradient m and y-intercept (0,c) is y = mx + c The Straight Line y b x a. (a,b).

5. The equation of a straight line with gradient m and passes through the point (a,b) is y - b = m(x - a) The general form of the equation is ax + by + c = 0 To be able to identify the gradient or y-intercept: 1.the equation of a straight line must be in the form y = mx + c 2.if the equation is not in that form change the subject of the formula Parallel lines have equal gradients and distinct y-intercepts A set points are Collinear if they all lie on a single straight line: 1.find the gradients of the lines joining pairs of points 2.equal gradients indicate parallel lines 3.a common point determines collinearity m = tan θ where θ is the angle between a line and the positive direction of the x-axis Lines with gradients m 1 & m 2 are perpendicular if m 1 × m 2 = -1

6. ✓ A -2 B -⅔ C 1.5 D 2 The line a has equation 3y + 2x = 6. What is the gradient of any line parallel to a? Vote for the correct answer now 3y + 2x = 6 3y = -2x + 6 Remember gradients of parallel lines are equal!!

7. y What is the equation of the line l 2 which passes through the point (8,-2) and is perpendicular to the line l 1 ? Step 1: Find the gradient of the line l 1 ? What is the angle given in degrees? What is the angle between l 1 and the positive direction of the x-axis? What is the gradient of the line l 1 ? 45° 135° m l1 = tan135° = -1 x l1l1 ?? Step 2: Find the equation of the line l 2 ? m l2 = 1 What is the gradient of l 2 ? Why? What is the equation of the line l 2 ? for perpendicular lines m l1 x m l2 = -1 y + 2 = 1(x – 8) y = x - 10

8. ✓ A 8.72 B C D What is the distance between the points (-2,5,3) and (4,- 1,1) as an exact value in its simplest form? Vote for the correct answer now

9. The points A(0,8), B(-4,0) and C(p,-4) are collinear. What is the value of p? What is the gradient of AB?m AB = m AB = m BC If A, B & C are collinear, what do we know about the gradients of AB & BC? So -4 = 2(p + 4) -4 = 2p + 8 -12 = 2p p = -6 What is an expression for the gradient of BC? m BC =

10. If the points M(a, b), N(c, d) and P(e, f) are the vertices of a triangle then the coordinates of the centroid are The centroid lies two thirds of the distance along each median measured from its vertex. A median of a triangle is a line from a vertex to the midpoint on the opposite side The medians of a triangle are concurrent and the point of intersection is called the centroid. Triangles: A Median A C B

11. An altitude of a triangle is a line from a vertex, which is perpendicular to the opposite side. Altitudes and Perpendicular Bisectors A C B The perpendicular bisector of a side in a triangle is the line which is perpendicular through the midpoint of the side. A B C The altitudes of a triangle are concurrent and the point of intersection is called the orthocentre The perpendicular bisectors of the sides of a triangle are concurrent and the point of intersection is called the circumcentre. You must know which is which, i.e. median, altitude and perpendicular bisector.

12. What is the equation of the line p, the median from C? What is a Median? What are the coordinates of R, the midpoint of AB? What is the gradient of CR? What is the equation of p (the median from C). What is the equation of the line q, the altitude from A? What is an Altitude? What is the gradient of BC? What is the gradient of the line perpendicular to BC? What is the equation of the line q (the altitude from A). What are the coordinates of the point of intersection of lines p and q? Using simultaneous equations with your answers to the previous parts of the question often helps. The Triangle ABC has vertices A(-1,6), B(-3,-2) and C(5,2). R(-2,2) m CR = 0, CR is a horizontal line y = 2 m BC = ½ m perp = -2 y - 6 = -2(x + 1) y = -2x + 4 2 = -2x + 4 -2 = -2x x = 1 The point of intersection of the lines p and q is (1,2) The line from C to the mid point of AB The line from A perpendicular to BC

13. The triangle with vertices A(8,6), B(9,-5) and C(-2,y) has centroid with coordinates (a, 1). What is the value of a? What is the Centroid? The centroid is the point of intersection of the medians = 5 What is the value of a? What is the value of y? 1 + y = 3 y = 2

14. The triangle ABC has vertices A(2,2), B(12,2) and C(8,6). What is the equation of l 1, the perpendicular bisector of AB? What is the mid point of AB? m AB = What is the gradient of the perpendicular bisector?undefined What is the gradient of AB? (7,2) Is the perpendicular bisector horizontal or vertical? vertical Find the equation of l 1 through the midpoint of AB ✓ A x = 7 B y = 2 Vote for the correct answer now x *(7,2)

15. Question Time If you have any questions on tonight’s session please ask The next session will be on the 29 th of October It will cover Polynomials & Quadratics Rayner will provide a link for you to give us feedback on tonight’s session If you need any help you can join me after this live on the ‘Ask a Tutor’ discussion board from 7-8pm You can ask questions on the forum at any time but I will be online most Thursdays from 7-8pm