SCHOLARHigher MathematicsHomework Session

Thursday 22nd October 6 - 7pm

You will need a pencil, paper and a calculator for some of the activities

My name isMargaret Ferguson

I am the SCHOLAR online tutor for Maths

The Straight Line

Tonight’s Session will cover

Given any two points (x1,y1) and (x2,y2)

•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

xa

.(a,b)

.(a,b)

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:

the equation of a straight line must be in the form y = mx + c

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:

find the gradients of the lines joining pairs of points

equal gradients indicate parallel lines

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 m1 & m2 are perpendicular if m1 × m2 = -1

✓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 = 63y = -2x + 6

Remember gradients of parallel lines are equal!!

y

What is the equation of the line l2 which passes through the point (8,-2) and is perpendicular to the line l1?

Step 1: Find the gradient of the line l1?What is the angle given in degrees?What is the angle between l1 and the positive direction of the x-axis?What is the gradient of the line l1?

45°

135°ml1 = tan135° = -1

x

l1

??

Step 2: Find the equation of the line l2?ml2 = 1What is the gradient of l2?

Why?What is the equation of the line l2?

for perpendicular lines ml1 x ml2 = -1y + 2 = 1(x – 8)y = x - 10

✓

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

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? mAB =

mAB = mBC

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 = 2pp = -6

What is an expression for the gradient of BC?

mBC =

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 MedianA

C

B

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.

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)mCR = 0, CR is a horizontal line

y = 2

mBC = ½ mperp = -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

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

= 5What is the value of a?

What is the value of y?

1 + y = 3

y = 2

The triangle ABC has vertices A(2,2), B(12,2) and C(8,6).What is the equation of l1, the perpendicular bisector of AB? What is the mid point of AB?

mAB =

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 l1 through the midpoint of AB

✓A x = 7

B y = 2

Vote for the correct answer now

x*(7,2)

Question Time

• If you have any questions on tonight’s session please ask• The next session will be on the 29th 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