Chapter 9_loci In Two Dimension.pdf

Math Form 2

CHAPTE R 9

9.1 The concept of Two-Dimensional Loci 9.2 Intersection of Two Loci

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9.1a

Locus Of A Moving Object

Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point X when a girl jogs along the park is a straight horizontal line. *

Locus of X *

Locus Of A Moving Object Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point Y on an apple when an apple drops from a tree is a straight vertical line. Locus of Y *

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Locus Of A Moving Object Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point P on the needle of a clock is a circle. Locus of P *

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Locus Of A Moving Object Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point Q on a ball when the ball is thrown is a curve. Locus of Q *

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Locus Of A Moving Object Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point B on a man when he runs around the football field is a rectangle. Locus of B *

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Locus Of A Moving Object Locus of a point in two-dimensions is the path produced when the point moves under certain condition(s).

Locus of a point R on the wheel of a car when it moves is a circle. *

Locus of R

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9.1b

1

The Locus of Points

The locus of points that are at a constant distance from a fixed point is a circle with radius equals to constant distance. Locus of points (circle) point

Fixed point

Construct a circle with radius equals to the constant distance.

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Constant distance (radius)

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The Locus of Points

The locus of points that are equidistant from two fixed point is the perpendicular bisector of the line joining the two fixed points. Line P

Locus of points (perpendicular bisector)

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

Construct a perpendicular bisector of the line that joins the two fixed points.

point

A l Fixed point

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Line P is the perpendicular bisector of AB

l

B Fixed point

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The Locus of Points

The locus of points that are at a constant distance from a straight line in a pair of parallel lines at the constant distance from the given straight line.

Locus of points (parallel lines)

Given straight line

Construct parallel lines at the constant distance from the given straight line.

Constant distance Constant distance

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Locus of points (parallel lines)

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The Locus of Points

The locus of points that are equidistant from two intersecting lines is the angle bisector of the angle formed by the two intersecting lines.

Construct bisectors of the angles formed by the intersecting lines.

Locus of points (angle bisector) xº xº

2 intersecting lines

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9.2

Intersecting of Two Loci

Intersecting of two loci in two dimensions is the point(s) which satisfy the conditions of the two loci. A

For example: CA is the locus of a moving point which is equidistant from CD and CB.

B

DB is the locus of a moving point which is equidistant from DA and DC

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D

C

 O is the intersecting of the two loci. *

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1. Draw a line PQ in 3cm length. It has finished already, good job! a.Construct locus X of a moving Locus X point which is 2 cm from the point Q. b.Construct locus Y of a moving point which is equidistant from the points P and Q. M Mark M as the point(s) of intersection of the two loci. 2 cm ll P

ll

Q

3 cm

M= intersection of the two loci M Locus Y

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2. Draw a square ABCD with sides 3 cm. Construct: a.P is the locus of a point moving in the square so that the point is 2 cm from B. b.Q is the locus of a point moving in the square so that the point is equidistant from AD and DC.

Locus Q

A

Mark X as the point(s) of intersection of the two loci.

3 cm

D

Locus P

x 3cm 2 cm Solution: Locus P is a quarter circle radius 2 cm with centre at B. Locus Q is the angle bisector of ADC, i.e, the line BD. B *

C

X = intersection of the two loci

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3. In the diagram, ABCD is a rectangle with sides 4 cm and 3 cm. Construct: a.P is the locus of a point moving in the rectangle so that it is 2 cm from AB. b.Q is the locus of a point moving in the rectangle so that it is equidistant from C and D.

A

4 cm

B

Locus Q

Mark X as the point of intersection of the two loci.

2 cm 3 cm

Locus P x

Solution: Locus P is a parallel line, 2 cm from AB. Locus Q is the perpendicular bisector of DC.

C

D

X = intersection of the two loci *

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4. Given that PQR is a triangle with sides 3 cm. Construct: a.M is the locus of a point moving in the triangle so that the point is 2 cm from Q. b.N is the locus of a point moving in the triangle so that it is equidistant from PR and QR.

R

Locus N

Locus M

Mark X as the point of intersection of the two loci. x 2 cm Solution: Locus M is an arc of radius 2 cm with center at Q. Locus N is the angle bisector of PRQ.

P

Q

X = intersection of the two loci *

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