Advertisements
Advertisements
प्रश्न
Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords, AB and AC, of the circle of length 6 cm and 5 cm respectively.
- Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction.
- Construct the locus of points, inside the circle, that are equidistant from AB and AC.
Advertisements
उत्तर १
- Draw PQ, the perpendicular bisector of chord AC. PQ is the required locus, which is the diameter of the circle.
Reason: We know each point on the perpendicular bisector of AB is equidistant from A and B. Also, the perpendicular bisector of a chord passes through the centre of the circle, and any chord passing through the centre of the circle is its diameter.
∴ PQ is the diameter of the circle. - Chords AB and AC intersect at M and N is a moving point such that LM = LN, where LM ⊥ AB and LN ⊥ AC
In right ΔALN and ΔALB
∠ANL = ∠ABL ...(90° each)
AL = AL ...(Common)
NL = BL ...[Given]
∴ ΔALN = ΔALB ...[R.H.S.]
Hence ∠MAL = ∠BAL ...c.p.c.t.
Thus, L lies on the bisector of ∠BAC.
Hence proved.
उत्तर २
Draw a circle of radius 4 cm whose center is O. Take a point A on the circumference of this circle.
With A as the center and a radius of 6 cm, draw an arc to cut the circumference at B. Join AB.
Then AB is the chord of the circle of length 6 cm.
With A as the center and a radius of 5 cm, draw another arc to cut the circumference at C. Join AC; then AC is the chord of the circle of length 5 cm.
With A as the center and a suitable radius, draw two arcs on opposite sides of AC.
With C as the center and the same radius, draw two arcs on opposite sides of AC to intersect the former arcs at P and Q.
Join PQ and produce to cut the circle at D and E.
Join DE. Then chord DE is the locus of points inside the circle that Ls equidistant from A and C.
As chord DE passes through the center O of the circle, it is a diameter. To prove the construction, take any point S inside the circle on DE.
APPEARS IN
संबंधित प्रश्न
Describe the locus of a point in space, which is always at a distance of 4 cm from a fixed point.
Describe the locus of a point P, so that:
AB2 = AP2 + BP2,
where A and B are two fixed points.
O is a fixed point. Point P moves along a fixed line AB. Q is a point on OP produced such that OP = PQ. Prove that the locus of point Q is a line parallel to AB.
Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB.
Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of lengths 6 cm and 5 cm respectively.
(i) Construct the locus of points, inside the circle, that are equidistant from A and C. prove your construction.
(ii) Construct the locus of points, inside the circle that are equidistant from AB and AC.
AB and CD are two intersecting lines. Find a point equidistant from AB and CD, and also at a distance of 1.8 cm from another given line EF.
Construct a rhombus ABCD whose diagonals AC and BD are 8 cm and 6 cm respectively. Find by construction a point P equidistant from AB and AD and also from C and D.
In given figure 1 ABCD is an arrowhead. AB = AD and BC = CD. Prove th at AC produced bisects BD at right angles at the point M
Draw and describe the lorus in the following cases:
The locus of points at a distance of 4 cm from a fixed line.
Using ruler and compasses construct:
(i) a triangle ABC in which AB = 5.5 cm, BC = 3.4 cm and CA = 4.9 cm.
(ii) the locus of point equidistant from A and C.
(iii) a circle touching AB at A and passing through C.
