Geometry (Grid & section formula)

1. Coordinate Grid & Distance Geometry Formula T- 1-855-694-8886 Email- [email protected] By iTutor.com

2. Grid Grid A pattern of horizontal and vertical lines, usually forming squares. Coordinate grid A grid used to locate a point by its distances from 2 intersecting straight lines A B C D E 1 2 3 4 5 What are the coordinates for the foot ball?

3. The Coordinate Plane In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales: X axis – A horizontal number line on a coordinate grid. Y axis - A vertical number line on a coordinate grid. 1 2 3 4 5 6 x 1 2 3 4 5 6 y

4. Coordinates Coordinates An ordered pair of numbers that give the location of a point on a grid. 1 2 3 4 5 6 1 2 3 4 5 6 (3,4)

5. How to Plot Ordered Pairs Step 1 – Always find the x value first, moving horizontally either right (positive) or left (negative). Step 2 – Starting from your new position find the y value by moving vertically either up (positive) or down (negative). (3, 4) 1 3 2 4 5 1 2 3 4 5 6 y 6 1 3 2 4 5 1 2 3 4 5 6 y 6 Step 1 Step 2 x x (3, 4)

6. Four Quadrants of Coordinate Grid Origin – The point where the axes cross is called the   origin  and is where both x and y are zero. On the x-axis, values to the right are positive and those to the left are negative.  On the y-axis, values above the origin are positive and those below are negative.

7. Four Quadrants of Coordinate Grid When the number lines are extended into the negative number lines you add 3 more quadrants to the coordinate grid. -2 -1 1 2 -3 3 -2 -1 1 2 -3 3 y x (+ , +) ( -, +) ( -, -) (+ , - ) 1 st Quadrant 2 nd Quadrant 3 rd Quadrant 4 th Quadrant

8. Four Quadrants The following relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies. If a point is in the 1st quadrant, then the point will be in the form (+, +), since the 1st quadrant is enclosed by the positive x - axis and the positive y- axis. If a point is in the 2nd quadrant, then the point will be in the form (–, +), since the 2nd quadrant is enclosed by the negative x - axis and the positive y - axis.

9. Four Quadrants If a point is in the 3rd quadrant, then the point will be in the form (–, –), since the 3rd quadrant is enclosed by the negative x - axis and the negative y – axis. I f a point is in the 4th quadrant, then the point will be in the form (+, –), since the 4th quadrant is enclosed by the positive x - axis and the negative y - axis x y (+, +) (–, +) (–, –) (+, –) I II III IV

10. Coordinate Geometry A system of geometry where the position of points on the plane is described using an ordered pair of numbers . The method of describing the location of points in this way was proposed by the French mathematician René Descartes . He proposed further that curves and lines could be described by equations using this technique, thus being the first to link algebra and geometry.  In honor of his work, the coordinates of a point are often referred to as its Cartesian coordinates, and the coordinate plane as the Cartesian Coordinate Plane. René Déscartes (1596 -1650)

11. Distance Formula The distance of a point from the y-axis is called its x-coordinate , or abscissa . The distance of a point from the x-axis is called its y-coordinate , or ordinate . The coordinates of a point on the x-axis are of the form (x, 0), and of a point on the y-axis are of the form (0, y).

12. Distance Formula Let us now find the distance between any two points P(x 1 , y 1 ) and Q(x 1 , y 2 ) Draw PR and QS  x-axis . A perpendicular from the point P on QS is drawn to meet it at the point T So, OR = x 1 , OS = x 2 , PR = PS = y 1 , QS = y 2 Then , PT = x 2 – x 1 , QT = y 2 – y 1 x Y P (x 1 , y 1 ) Q(x 2 , y 2 ) T R S O

13. Distance Formula Now, applying the Pythagoras theorem in ΔPTQ , we get Therefore which is called the distance formula.

14. Section Formula Consider any two points A(x 1 , y 1 ) and B(x 1 ,y 2 ) and assume that P (x, y) divides AB internally in the ratio m 1 : m 2 i.e. Draw AR , PS and BT  x-axis . Draw AQ and PC parallel to the x-axis. Then, by the AA similarity criterion , x Y A (x 1 , y 1 ) B(x 2 , y 2 ) P (x , y) R S O T m 1 m 2 Q C

15. Section Formula Δ PAQ ~ Δ BPC ---------------- (1) Now, AQ = RS = OS – OR = x– x 1 PC = ST = OT – OS = x 2 – x PQ = PS – QS = PS – AR = y– y 1 BC = BT– CT = BT – PS = y 2 – y Substituting these values in (1), we get

16. Section Formula For x - coordinate Taking or or or

17. Section Formula For y – coordinate Taking or or or

18. Section Formula So, the coordinates of the point P (x, y) which divides the line segment joining the points A (x 1 , y 1 ) and B (x 2 , y 2 ) , internally, in the ratio m1: m2 are This is known as the section formula .

19. Mid- Point The mid-point of a line segment divides the line segment in the ratio 1 : 1 . Therefore, the coordinates of the mid-point P of the join of the points A (x 1 , y 1 ) and B (x 2 , y 2 ) is From section formula

20. Area of a Triangle Let ABC be any triangle whose vertices are A (x 1 , y 1 ), B (x 2 , y 2 ) and C (x 3 , y 3 ) . Draw AP, BQ and CR perpendiculars from A , B and C , respectively, to the x-axis . Clearly ABQP , APRC and BQRC are all trapezium, Now, from figure QP = (x 2 – x 1 ) PR = (x 3 – x 1 ) QR = (x 3 – x 2 ) x Y A (x 1 , y 1 ) B(x 2 , y 2 ) C (x 3 , y 3 ) P Q O R

21. Area of a Triangle Area of Δ ABC = Area of trapezium ABQP + Area of trapezium BQRC – Area of trapezium APRC . We also know that , Area of trapezium = Therefore, Area of Δ ABC = Area of Δ ABC

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