Slope and Inclination of a Straight Line

How about the steepness of these two lines? You are right! In fact, in coordinate geometry, we use or to describe the steepness of a straight line. How about the steepness of these two lines? Let’s consider the two paths below. Which path is steeper? slope inclination x y Straight line B Straight line A Path A Path B It seems that straight line B is steeper. Of course, path B is steeper.

Slope of a Straight Line The slope of a straight line is the ratio of the vertical change to the horizontal change between any two points on the straight line, horizontal change vertical change x y A B i.e. line straight a of slope = change vertical horizontal change

Consider a straight line L passing through A(x1, y1) and B(x2, y2), where x1  x2. Coordinates of C = (x2, y1) y L B( , ) x2 y2 change horizontal vertical line straight a of Slope = vertical change x A( , ) x1 y1 C ( , ) x2 y1 1 2 y - = horizontal change 1 2 x -

If we use the letter m to represent the slope of the straight line L, then x y A(x1, y1) B(x2, y2) L 1 2 x y m - = 2 1 x y + - 2 1 x y - 2 1 x (x y - (y ) or m = 2 1 x y m -  1 2 x y m -  Note: and

Let’s find the slope of AB. y x A(–1, –1) B(4, 3) 1 2 - x y of Slope = AB 1) ( 3 - = (x1, y1) = (1, 1) (x2, y2)= (4, 3) 1) ( 4 - 5 4 =

Let’s find the slope of AB. x y A(–1, –1) B(4, 3) Alternatively, 2 1 - x y of slope = AB 3 1 - = (x1, y1) = (1, 1) (x2, y2)= (4, 3) 4 1 - 5 4 =

Follow-up question In each of the following, find the slope of the straight line passing through the two given points. (a) A(2, 4) and B(3, –2) (b) C(1, 1) and D(3, 5) Solution 3 2 ) ( 4 of Slope - = AB 1 3 5 of Slope - = CD (a) (b) 6 - = 2 = The slopes of AB and CD are in opposite sign. What does this mean?

In fact, for straight lines sloping upwards from left to right, their slopes are positive. for straight lines sloping downwards from left to right, their slopes are negative. x y Slope = 2 Slope = 1 Slope = 0.5 x y Slope = –0.5 Slope = –1 Slope = –2 Slope Slope > 0 < 0

The greater the value of the slope, the steeper is the straight line. In fact, x y Slope = 2 Slope = 1 Slope = 0.5 for straight lines sloping upwards from left to right, their slopes are positive. x y Slope = –0.5 Slope = –1 Slope = –2 for straight lines sloping downwards from left to right, their slopes are negative. The steepest line The steepest line Slope > 0 Slope < 0 The greater the numerical value of the slope, the steeper is the straight line. The greater the value of the slope, the steeper is the straight line. 2 2 1 0.5 > 1 > 0.5 > >

What are the slopes of a horizontal line and a vertical line?

The slope of a horizontal line is . x y A(x1, y1) B(x2, y1) For a line that is parallel to the x-axis, of Slope 1 2 - = x y AB = 2. The slope of a vertical line is . undefined x y D(x1, y1) C(x1, y2) For a line that is parallel to the y-axis, of Slope 1 2 - = x y CD 1 2 - = y  It is meaningless to divide a number by 0.

Follow-up question On the rectangular coordinate plane as shown, L1, L2, L3 and L4 are four straight lines. Given that their slopes are 0, 0.5, 1 and 2 (not in the corresponding order), determine the slopes of each line according to their steepness. x y L1 Straight Line L1 L2 L3 L4 Slope L2 L3 2 1 0.5 L4 L1 and L2 are sloping upwards from left to right and L1 is steeper. L4 is sloping downwards from left to right. L3 is a horizontal line.

Inclination We can also describe the steepness of a straight line by its inclination. y  is the angle that the straight line L makes with the positive x-axis (measured anti-clockwise from the x-axis to L) Straight line L  x positive x-axis  is called the inclination of L. Note: For 0 <  < 90, when  increases, the steepness of L also increases.

Is there any relationship between the inclination of a straight line and its slope?

Draw a horizontal line from A and Consider a straight line L passing through A and B with inclination  . y x A B L a Draw a horizontal line from A and C  a vertical line from B. They intersect at C. Let BAC = a.

Consider a straight line L passing through A and B with inclination  . y L B A a C  AC BC AC BC x L = of Slope L = of Slope  = a ∵   and a are corresponding angles. ∴ a = tan tan q tan q Note that ACB = 90. AC BC =  By the definition of tangent ratio ∴ tan  =

If the inclination of a straight line L is 50, slope of L = tan 50 The relationship between the inclination  and the slope of a straight line L is slope of L = tan  For example: If the inclination of a straight line L is 50, slope of L = tan 50 = 1.19 (cor. to 3 sig. fig.) y L 50 x

Let’s find the inclination  of L. y L Slope of L = tan  ∵ slope = ∴ = tan  60 x  60 = q

Follow-up question 1. Given that the inclination of a straight line L is 35, find the slope of L correct to 3 significant figures. Solution 1. Slope of L = tan 35 fig.) sig. 3 to (cor. 700 . =

Follow-up question 2. Given that the slope of a straight line L is 2, find the inclination  of L correct to the nearest degree. Solution 2. Slope of L = tan  2 = tan  ∴  63 = q (cor. to the nearest degree)