Presented by: LEUNG Suk Yee LEUNG Wing Yan HUI Hon Yin LED 3120B PowerPoint Presentation

Target Audience : F.2 Students Prerequisite knowledge : learnt about the distance between two points on the same horizontal line or on the same vertical line in a rectangle coordinate plane. The use of slides : source of learning

LEUNG SUK YEE : Data Collection & Slide Production LEUNG WING YAN : Data Collection & Slide Production HUI Hon Yin: Slide Production & Layout Design

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1) Distance between any TWO points 2) Slope of a STRAIGHT line 3) PARALLEL lines 4) PERPENDICULAR lines 5) MC Questions

1.1 Distance between two points on the same horizontal line y x0 The Distance of AB: AB = 4-1 = 3 units A(1,1)B(4,1)

1.2 Distance between two points on the same vertical line y x0 Distance of AC: AC = 5-1 = 4 units C(1,5) A(1,1)

1.3 Distance between any two points y x0 Then, do you know the distance between these two points ? C(1,5) B(4,1)

i) Draw a vertical line from A through C ii) Draw a horizontal line from A through B iii) Since ABC is right-angled triangle, do you remember that we can use the Pythagoras’ theorem to calculate the hypotenuse BC? y x0 C(1,5) B(4,1) A(1,1)

By the Pythagoras’ theorem,we have BC 2 =AB 2 +AC 2 = =9 +16 =25 ∴ BC =5 y x0 C(1,5) B(4,1) A(1,1) Why???

From the figure, we have AB = x 1 - x 2 and AC = y 1 – y 2 By Pythagoras’ theorem, we have BC 2 = AB 2 + AC 2 = (x 1 - x 2 ) 2 + (y 1 – y 2 ) Distance Formula y x0 C(x 2,y 1 ) B(x 1,y 2 ) A(x 2,y 2 )

BC 2 = (x 1 - x 2 ) 2 + (y 1 - y 2 ) 2 = (4-1) 2 + (2-6) 2 = (-4) 2 = 25 ∴ BC=5 y x0 C(1,5) B(4,1)

C(-1,4) A(-2,0) B(2,-1) y x Find the perimeter of ABC correct to 1 decimal place. Solution : Example 1 Time Remain 5 sec Time Remain 3 sec Time Remain 2 sec Time Remain 1 sec Time Remain 4 sec AB 2 =(-2 -2) (-1) 2 =17 BC 2 =2-(-1) 2 + (-1- 4) 2 =34 AC 2 = -2 - (-1) 2 + (0-4) 2 =17 Perimeter of ABC =AB + BC +AC = =14.1(correct to 1d.p.)

Lines AB and CD represent two steep roads. Which road is steeper? Obviously road CD is steeper than road AB. But, can you tell the steepness of the roads? 2.1 Definition of Slope A B D C

We can determine in this way Difference in x-coordinates Difference in y-coordinates (x 1, y 2) (x 1, y 1) (x 2,y 2) A(x, y ) and B (x, y ) are two points on a straight line. Difference in y- coordinates =y 1 -y 2 Difference in corresponding x- coordinates =x 1 -x 2 Hence, Slope of a line = y 1 -y 2 / x 1 -x 2

... P(2,3) Q (12,9) R (7,9) Example 2 Answer: Slope of PQ = 3-9 / 2-12 = -6 / -10 = 3/5 Slope of PR = 3-9 / 2-7 = -6 / -5 = 6/5 Find the slope of the lines PQ & PR Time Remain 5 sec Time Remain 4 sec Time Remain 3 sec Time Remain 2 sec Time Remain 1 sec

Example 3 Answer: Slope of XY = 1-10 / 10-4 = -9 / 6 = -3 / 2 Slope of XZ = 1-7 / 10-2 = -6 / 8 = -3 / 4 Find the slope of the lines XY & XZ.. Z(2,7) Y (4,10) X (10,1) Time Remain 5 sec Time Remain 4 sec Time Remain 3 sec Time Remain 2 sec Time Remain 1 sec

From the above two example, we can conclude that: (1) when a straight line rises from left to right, the slope of the line is a positive. (2) when a straight line falls from left to right, the slope of the line is negative. Line with positive slope Line with negative slope 2.2 The Characteristics of slope

There is a pair of parallel lines. What’s the different between there slope? Slopes of Parallel Lines

Slope of AB = 1-3 / 1-5 = -2 / -4 =1 / 5 A(1,1) B(5,3) C(-1,1) D(2,2.5) Slope of CD = / = (-3 / 2) / -3 = 1 / 2 So, Slope of AB = Slope of CD =1 / 2

In general, If two lines are parallel, then their slopes are equal. Conversely, If two lines have same slope, then they are parallel.

A(3, -1) B(4,3) C(-2,-2) D(-3/2,0) Show that two straight lines AB and CD are parallel. Answer: Slope of AB = (-1-3)/3-4 = -4/-1 = 4 Slope of CD = (-2-0)/-2-(-3/2) = -2/(-1/2) = 4 Example 4 Since slope of AB = slope of CD, so AB and CD are parallel lines. Time Remain 5 sec Time Remain 4 sec Time Remain 3 sec Time Remain 2 sec Time Remain 1 sec

D(1,6) B(6,5) C(5,0) A(0,1) Slopes of Perpendicular Lines There is a pair of perpendicular lines. What’s the relationship between there slope?

D(1,6) B(6,5) C(5,0) A(0,1) Slope of AB = 5-1 / 6-0 = 4 / 6 = 2/3 Slope of CD = 6-0 / 1-5 = 6 / -4 = -3/2 Slope of AB x Slope of CD = 2/3 x (-3/2) =-1

In general, If two lines are perpendicular, then the product of their slopes are equal to -1. Conversely, If the product of their slopes are equal to -1,then they are perpendicular.

Example 5 Show that two straight lines AB and CD are perpendicular. A(0,3) B(-2,0) C(-6,4) D(0,0) Answer: Slope of AB = (3-0)/0-(-2) = 3/2 Slope of CD = (4-0)/(-6)-0 = -4/6 = -2/3 ∵ Slope of AB x slope of CD =3/2 x (-2/3) =-1 ∴ AB and CD are perpendicular. Time Remain 5 sec Time Remain 4 sec Time Remain 3 sec Time Remain 2 sec Time Remain 1 sec

1. Find the perimeter of a square if X(- 3,8) and Y(2,-4) are two adjacent vertices of the square. A. 13 B. 52 C. 169

2. In the figure, which line has the greatest slope? A. L1 B. L2 C. L3 x y L3 L2 L1

.... (4,3) (3,n) (2,5) (1,0) A C B D x y 3. In the figure, AB is parallel to CD. What is the value of n? C. 8 B. -1 A. -2

4. Given two points M(2,0) and N(5,1). If P(3,b) is a point such that MP=MN, find b.. A. 3 only B. 2 or -2 C. 3 or -3

.. D (a,b) (-1,-2) C A B x y 5. In the figure, AB is perpendicular to CD. (a,b) and (-1,-2) are two points on CD. If the slope of AB is –1/2, express a in terms of b. A. a=b/2 B. a=-b/2 C. a=2b+3

END Produced by Leung Suk Yee Leung Wing Yan Hui Hon Yin

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RIGHT Well done, You have finish all the question!

13 is only the length of one side of the square, not the perimeter of the square. Wrong Don’t give up! Let’s try it again.

169 is the area of the square, not the perimeter of the square Wrong Don’t give up! Let’s try it again.

L1 is parallel to the x-axis, there is no difference in y-coordinates, therefore slope of L1 is Slope of L1 = 0/x 1 -x 2 = 0 Wrong Don’t give up! Let’s try it again.

L2 falls from left to right, the slope must be negative. Wrong Don’t give up! Let’s try it again.

You may have some problems in using the slope formula. Slope formula = y 1 -y 2 / x 1 -x 2 Are you sure that you have put x 1,x 2,y 1 and y 2 in the right place of the formula ? Wrong Don’t give up! Let’s try it again.

In fact, 9 can be the square of both 3 and -3. Wrong Don’t give up! Let’s try it again.

The situation is MP =MN, not MP = PN. Wrong Don’t give up! Let’s try it again.

You may make some mistakes in the process of calculation. Please be more careful. Wrong Don’t give up! Let’s try it again.

The slope formula is y 1 -y 2 / x 1 -x 2, not x 1 -x 2 / y 1 -y 2 Wrong Don’t give up! Let’s try it again.