MODULE - 8 ANALYTICAL GEOMETRY

Represent geometric figures on a Cartesian plane and derive and apply, for any two points a formula for calculating: (a) distance between two points (b) the gradient of the line segment joining the points (c) the co – ordinates of the mid – point of the line segment joining the points (LO 3 AS 3) Through investigations, produce conjectures and generalizations related to triangles, quadrilaterals and other polygons, and attempt to validate, justify, explain and prove them, using any logical method. (LO 3 AS 2)

Analytical Geometry deals with the study of geometry using the Cartesian Number Plane. It is an algebraic approach to the study of Geometry. In this module, we will deal with the following topics: The distance formula The midpoint formula The gradient of a line segment joining two point.

THE DISTANCE FORMULA The distance formula enables us to calculate the distance between two given points on the Cartesian Number Plane. It actually calculates the length of the line segment joining these two points. The following activity will be used to derive the distance formula.

Let’s investigate. 1. Answer the following questions based on the diagram below. (a) Calculate the value of (b) Calculate the length of BC (c) Calculate the value of (d) Calculate the length of AC (e) Now use the theorem of Pythagoras to calculate the length of AB

2. Answer the following questions based on the diagram below.

(a) Write down the length of BC in terms of and (b) Write down the length of AC in terms of and (c) Now use the theorem of Pythagoras to determine the length of AB in terms of , , and . The formula to calculate the length of a line segment between two points is given by the following formula called the distance formula: or Where the two points joining line segment AB are A ( ) and B ( )

Example 1 Calculate the length of the line segments in the following diagrams.

(A)

(b)

Example 2 In the diagram below, the coordinates of are A (2; 3), B (5; 7) and C (- 2; 6). (a) Show that is an isosceles triangle. (b) Show that is a right-angled triangle.

(a) We can show that is an isosceles triangle by proving that AB = AC.

We can show that is a right-angled triangle by proving the theorem of Pythagoras statement From the previous part of the example, we proved that AB =5 and AC =5. All we must now to is calculate the length of BC:

Now, and Therefore A = 90° and is a right-angled triangle.

EXERCISE 1 1. Calculate the length of the line segments in the following diagram.

2. In the diagram below, two triangles have been drawn. (a) Show that is an isosceles triangle. (b) Show that is a right-angled triangle. (c) Show that is a scalene triangle.

3. (a) Determine the length of the perimeter of a triangle whose vertices are L (- 1; 2), M (l; 6) and N (4; 0). (b) Show that the points A (3; 6) and B (- l; 4) are equidistant from the point C (2; 3).

Three points A, B and C are said to Three points A, B and C are said to be collinear if they lie on the same straight line. A property of collinear points is that AB + BC = AC. By using this property, show that the following points are collinear: A (- 1; 0), B (1; 2) and C (3; 4)

(Hint: You will need to solve a quadratic equation). 5. If the length of the line segment joining the points A (1; 2) and B(x; 6) is equal to 5 units, determine the two possible values of x. (Hint: You will need to solve a quadratic equation).

THE MIDPOINT FORMULA The midpoint of a line segment is the halfway mark on the line segment. Let us try!!

1. (a) Write down the coordinates of M__________ (b) Calculate: (c) What do you notice? (d) Calculate: (e) What do you notice?

2. (a) Write down the coordinates of M_______________________ (b) Calculate: (c) What do you notice? (d) Calculate: (e) What do you notice? The formula to calculate the midpoint of a line segment between two points is given by the formula: where the two points joining line segment AB are A( and B(

Example 1 Calculate the midpoints of the following line segments: Midpoint of AB: A (l; 3) and B (5; 7) Midpoint of CD: C (- 5; 6) and D (- 1; 2)

Example 2 Determine the value of x and y if M (5; 2) is the midpoint of the line segment joining the points A (x; 1) and B (8; y).

EXERCISE 2 Determine the midpoints of the following line segments:

2. (a). If M (- 3; 2) is the midpoint of the line 2. (a) If M (- 3; 2) is the midpoint of the line joining the points A(x; 1) and B (- l; y), calculate the value of x and y. (b) If M (- 1; 7) is the midpoint of the line joining the points A(x; 6) and B (2; y), calculate the value of x and y. (c) If M (- 1; - 5) is the midpoint of the line joining the points A(x; y) and B (- 6; - 3), calculate the value of x and y.

THE GRADIENT OF A LINE SEGMENT We will now focus on a concept referred to as the gradient of a line between any two points on the line. Gradient is really a change in y-values in relation to a corresponding change in the x-values. This rate of change of the one variable with respect to the other can be expressed as a ratio:

change in y - values change in x - values To determine the gradient of a line between two points, we can use the idea of the change in y - values as a vertical movement from a point on the line followed by the corresponding change in x - values as a horizontal movement.

Example 1 Consider the line segment AB in the diagram below. The gradient of the line segment AB can be seen as a vertical movement of 4 units upward from A followed by a horizontal movement of 8 units to the right landing up at B.

Gradient of AB = change in y - values change in x - values = vertical movement upwards horizontal movement right Here the line slopes to the right and the gradient is positive.

Example 2 Consider the line segment AB in the diagram below The gradient of the line segment AB can be seen as a vertical movement of 3 units downward from A followed by a horizontal movement of 9 units to the right landing up at B..

Gradient of AB = change in y – valueas change in x – values = vertical movement downwards horizontal movement right = - 3 +9 = Here the line slopes to the left and the gradient is negative.

A formula to calculate the gradient of a line between two points Let’s try!!!! Consider the following line segments:

(a) Complete the following: Gradient AB = What do you notice (b) Complete the following: Gradient CD = What do you notice?

A formula to calculate the gradient of a line joining two points. The gradient of line where A ( and B ( ) are two points on the line.

Example 3 Calculate the gradients of the following lines using the formula for gradient.

Gradient

Let’s Try!!!! (a) Calculate the gradients of the following line segments: Gradient AB = Gradient GH = Gradient CD = Gradient JK = Gradient LM =

(b) What do you notice about lines AB, CD and EF? (c) What do you notice about lines GH, JK and LM?

Important conclusion about parallel lines: If two or more lines are parallel, then these lines have equal gradients.

Here we go!!!! Consider the following line segments with AB perpendicular to CD and EF perpendicular to GH.

(a) Determine: Gradient AB = (Gradient AB) x (Gradient CD) = Gradient CD = Gradient EF = (Gradient EF) x (Gradient GH) = Gradient GH = (b) What do you notice about the product of the gradients of perpendicular line segments?

Important conclusion about perpendicular lines: If two are perpendicular, then the product of their gradients equals - 1.

The gradient of horizontal and vertical line segments Example Determine the gradient of the line joining the points: (a) A ( - 2; 3) and B (1; 3) Gradient AB = = Here the line is horizontal and has a gradient of zero

A (4; 3) and B (4;- 5) Gradient AB = undefined It is clear from the diagram below that the line is vertical.

The gradient is undefined. From the previous example, two extremely important facts emerge: The gradient of a horizontal line is always zero. The gradient of a vertical line is always undefined.

EXERCISE 3 1. Calculate the gradients of the following line segments:

2. Calculate the gradient of the lines joining the following points. (a) A (l; 3) and B (5; 7) (b) A (l; 3) and B (- 5; -7) (c) A (- l; -3) and B (5; 7) (d) A (- l; 3) and B (5; -7) (e) A (1; -3) and B (- 5; 7) (f) A (- l; -3) and B (- 5; -7)

3. By calculating the gradients of the following line segments, Determine whether the line segments are parallel, perpendicular or neither (a) AB if A (-l; -3) and B (2; 1) are points on the line AB and CD if C (4; -1) and D (7; 3) are points on line CD. (b) AB if A (1; -3) and B (2; 1) are points on the line AB and CD if (c) AB if A (l; -3) and B (2; 1) are points on the line AB and CD if C (- 3; 1) and D (l; 0) are points on line CD.

4. (a) Line segment AB is parallel to line segment CD. A (- 5; - 1) and B (- 3; a) are points on AB. C (- 4; - 3) and D (- 1; 3) are points on CD. Calculate the value of a. (b) Line segment AB is perpendicular to line segment CD. A (- 5; 2) and B (b; -1) are points on AB. C (- 4; -3) and D (-; 3) are points on CD. Calculate the value of b.

ASSESSMENT TASK This assessment task investigates properties of a triangle involving a line joining the midpoints of two sides of the triangle. This investigation makes use of Analytical Geometry, Transformation Geometry and some Euclidean Geometry.

In the diagram below, four triangles are drawn In the diagram below, four triangles are drawn. In each triangle, a line joining the midpoints of two sides is drawn. Answer the following questions based on the diagram.

1. (a). Calculate, using analytical methods, the 1. (a) Calculate, using analytical methods, the coordinates of M, the midpoint of side AB of . Indicate point M on the diagram. (b) Calculate, using analytical methods, the coordinates of N, the midpoint of side AC of. Indicate point N on the diagram. (c) Calculate, using analytical methods, the gradient of line MN. (d) Calculate, using analytical methods, the gradient of line BC. (e) What do you notice about line MN and BC? (f) Calculate, using analytical methods, the length of line MN. (g) Calculate, using analytical methods, the length of line BC.

2. (a). Calculate, using analytical methods, the. coordinates of P 2. (a) Calculate, using analytical methods, the coordinates of P. the midpoint of side DE of. Indicate point P on the diagram. (b) Calculate, using analytical methods, the coordinates of Q, the midpoint of side DF of. Indicate point Q on the diagram. (c) Calculate, using analytical methods, the gradient of line PQ. (d) Calculate, using analytical methods, the gradient of line EF. (e) What do you notice about line PQ and EF? (f) Calculate, using analytical methods, the length of line PQ. (g) Calculate, using analytical methods, the length of line EF. (h) What do you notice about the lengths of PQ and EF?

3. (a). Calculate, using analytical methods, the 3. (a) Calculate, using analytical methods, the coordinates of R, the midpoint of side GH of Indicate point R on the diagram. (b) Calculate, using analytical methods, the coordinates of S, the midpoint of side GI of Indicate point S on the diagram. (c) Calculate, using analytical methods, the gradient of line RS. (d) Calculate, using analytical methods, the gradient of line HI. (e) What do you notice about line RS and HI? (f) Calculate, using analytical methods, the length of line RS. (g) Calculate, using analytical methods, the length of line HI. (h) What do you notice about the lengths of RS and HI?

4. (a). Calculate, using analytical methods, the 4. (a) Calculate, using analytical methods, the coordinates of T, the midpoint of side LJ of Indicate point T on the diagram. (b) Calculate, using analytical methods, the coordinates of W, the midpoint of side KJ of Indicate point W on the diagram. (c) Calculate, using analytical methods, the gradient of line TW. (d) Calculate, using analytical methods, the gradient of line LK. (e) What do you notice about line TW and LK? (f) Calculate, using analytical methods, the length of line TW. (g) Calculate, using analytical methods, the length of line LK. (h) What do you notice about the lengths of TW and LK?

With reference to the triangle, write down a conjecture about the line joining the midpoints of two sides of a triangle. You will now be required to prove your conjecture.

6. Using the diagram provided below, reflect. about the dotted 6. Using the diagram provided below, reflect about the dotted vertical line. Draw the image. Then reflect the newly formed figure about the dotted horizontal line. Draw the image and call it L. It might be helpful to use tracing paper to do this. Fill in the equal sides and angles of the triangles. You may assume that line DE//BC.

7. (a) Prove, using congruency, that (b) Write down the properties of a parallelogram relating to its sides. (c) Prove that DBCF is a parallelogram. (d) Now prove your conjecture relating to the length of DE in relation to BC.