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9 LOCI IN TWO DIMENSIONS Prepared by : Rosila Othman
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Locus is the path of a moving point or a point or set of points that satisfies given conditions. 9.1 TWO DIMENSIONAL LOCI A figure of ‘8’ A circle
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circle vertical line pentagon square triangle curve / arc
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Describe and sketch the locus of the moving point <ul><li>The tip of a minute hand rotating on the face of a clock. </li></ul><ul><li>A circle </li></ul><ul><li>A stone is dropped from the first floor of a building. </li></ul><ul><li>A vertical line </li></ul><ul><li>The Earth revolves round the sun. </li></ul><ul><li>An ellipse / a oval circle </li></ul>
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<ul><li>A swinging pendulum. </li></ul><ul><li>An arc </li></ul><ul><li>The centre of the wheel of a moving vehicle on the road. </li></ul><ul><li>A horizontal straight line </li></ul><ul><li>A competitor running in a 400 m race in the field. </li></ul><ul><li>An oval </li></ul>
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Exercise: <ul><li>9.1A Question 2 </li></ul><ul><li>9.1B All </li></ul>
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The locus of a moving point P that is at a constant distance from a fixed point O is a circle with centre O . Locus of P The locus of P is a circle with radius OP and centre O . O P
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The locus of a moving point R equidistant from two fixed points A and B is the perpendicular bisector of the line AB . The locus of R is the perpendicular bisector of AB . || || A B Locus of R
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The locus of a moving point that is a constant distance from a straight line AB are two straight lines that are parallel to AB . The locus are two lines ST and UV that are parallel to AB . A B U V = = Locus S T
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The locus of a moving point that is at equidistant from two intersecting lines AB and CD is a pair of straight lines which bisect the angles between the two intersecting lines. The locus are two straight lines PQ and RS which bisect the angles between the two intersecting lines. A B C D P Q R S Locus
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Determine the locus of the points which satisfy the given condition A circle with centre O and a radius 6 cm. Two straight lines parallel to AB and 3 cm from line AB. Two angle bisectors. The perpendicular bisector of the line EF. A point P moves such that it is equidistant from the point E and F. A point P moves such that it is equidistant from two intersecting line AB and CD. A point P moves such that it is 3 cm from the line AB. A point P moves at a distance of 6 cm from a fixed point O.
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Constructing the locus <ul><li>To construct the locus : </li></ul><ul><li>Describe or sketch the locus. </li></ul><ul><li>Decide on a suitable scale. </li></ul><ul><li>Construct the locus accurately. </li></ul>
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A circle with pupil B as the centre and a radius of 1.5 m 1.5 m Locus of pupil A Step 1: Describe or sketch the locus. Step 2: Decide on a suitable scale. Step 3: Construct the locus accurately. 1 cm represent 1 m. <ul><li>Place a pair of compasses on a ruler to measure a distance of 1.5 cm. </li></ul><ul><li>With the point pupil B as centre, draw an arc 1.5 cm from B to form a circle. </li></ul><ul><li>This is the locus of pupil A. </li></ul>
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Perpendicular bisector of line XY || || Locus of S Step 1: Describe or sketch the locus. Step 2: Decide on a suitable scale. Step 3: Construct the locus accurately. 1 cm represent 1 cm. <ul><li>Set your compasses to a length more than half of XY. Place the point of your compasses at X and draw an arc above and below the line. </li></ul><ul><li>With the same length, place the point of your compasses at Y and draw two arcs to intersect the first two arcs at A and B. </li></ul><ul><li>Draw a line through A and B. This is the locus of S. </li></ul>A B
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Two parallel lines at a constant distance of 1.8 cm from XY Locus of Z 1.8 cm 1.8 cm Step 1: Describe or sketch the locus. Step 2: Decide on a suitable scale. Step 3: Construct the locus accurately. 1 cm represent 1 cm. <ul><li>Mark a point A on the line XY. </li></ul><ul><li>Construct perpendicular bisectors to the line segment XA and AY. Mark the points of the intersection of the perpendiculars with line XY as B and C. </li></ul><ul><li>Set your compasses to a length of 1.8 cm. Place the point of your compasses at B and draw an arc on the perpendicular above and below the line. Repeat with the point of your compasses at C. </li></ul><ul><li>Draw a line 1.8 cm marks in step 3. This is the locus of Z. </li></ul>B C Locus of Z A
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Two angle bisectors of the angles formed by the line PQ and RS Locus of C Step 1: Describe or sketch the locus. Step 2: Decide on a suitable scale. Step 3: Construct the locus accurately. 1 cm represent 1 cm. <ul><li>Set a pair of compasses to about half of the length of OP. Place the point of your compasses at O and draw arcs to cut line OP and OR at A and B respectively. </li></ul><ul><li>Place the point of the compasses at A and then at B to draw two arcs that intersect. </li></ul><ul><li>Draw a line through O and the point where the arcs intersect. This line is the bisector of POB and SOQ. </li></ul><ul><li>Use the step 1, 2 and 3 as a guide to draw the bisector of POS and ROQ. The bisector of the angles is the locus of C. </li></ul>O A B Locus of C
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|| || Locus of P Locus of Q Locus of W 1 cm 1 cm B C Locus of W A
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The intersection of two loci is the point or points that satisfy the conditions of the two loci. The points of intersection of two loci that is (a) equidistant from A and B , (b) a constant distance from A . The points X and Y are the points of intersection of the two loci. 9.2 INTERSECTION OF TWO LOCI X Y Equidistant from A and B . A constant distance from A . A × B ×
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<ul><li>Construct a straight line XY of length 2.4 cm. Then construct the locus of </li></ul><ul><li>point P such </li></ul><ul><li>that it is always </li></ul><ul><li>1.5 cm from X . </li></ul><ul><li>point Q that is </li></ul><ul><li>equidistant </li></ul><ul><li>from X and Y . </li></ul><ul><li>Mark the point of intersection as A and B . </li></ul>2.4 cm Locus of Q Locus of P A B X Y
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<ul><li>Draw an equilateral triangle ABC with sides of length 3 cm. Then, construct the locus of point that is </li></ul><ul><ul><ul><li>equidistant </li></ul></ul></ul><ul><ul><ul><li>from A and </li></ul></ul></ul><ul><ul><ul><li>B . </li></ul></ul></ul><ul><ul><ul><li>2 cm from B . </li></ul></ul></ul><ul><li>Mark the point of intersection as D and E . </li></ul>3 cm C 3 cm 3 cm D E A B
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