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Standard X

Mathematics

Locus of the Points Equidistant From Two Given Points

Ruler and com...

Question

Ruler and compasses may be used in this question.All construction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment.

(i) Construct a $△ A B C$ ,in which BC=6 cm,AB=9 cm and angle ABC= $60^{o}$ .

(ii) Construct the locus of all points inside triangle ABC, which are equidistant from B and C.

(iii) Construct the locus of the vertices of the triangles with BC as base and which are equal in area to triangle ABC.

(iv) Mark the point Q, in your construction.which would make $△ Q B C$ equal in are to $△ A B C$ , and isosceles.

(v) Measure and record the length of CQ.

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Solution

Steps of Construction:

(i) Draw a line segment BC = 6 cm.

(ii) At B, draw a ray BX making an angle 60 degree and cut off BA=9 cm.

(iii) Join AC. ABC is the required triangle.

(iv) Draw perpendicular bisector of BC which intersects BA in M, then any point on LM is equidistant from B and C.

(v) Through A, draw a line m || BC.

(vi) The perpendicular bisector of BC and the parallel line m intersect each other at Q.

(vii) Then triangle QBC is equal in area to triangle ABC. m is the locus of all points through which any triangle with base BC will be equal in area of triangle ABC.

On measuring CQ = 8.4 cm.

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Similar questions

Construct a $Δ$ ABC, in which BC = 6 cm, AB = 9 cm and $∠$ ABC = 60 $^{\circ}$

i. Construct the locus of all points inside $Δ$ ABC, which are equidistant from B and C.

ii. Construct the locus of the vertices of the triangle with BC as base, which are equal in area to triangle ABC.

iii. Mark the point Q in your construction, which would make $Δ$ QBC equal in area to $Δ$ ABC, and isosceles.

iv. Measure and record the length of CQ.

Construct a $Δ$ ABC, in which BC = 6 cm, AB = 9 cm and $∠$ ABC = 60 $^{\circ}$

i. Construct the locus of all points inside $Δ$ ABC, which are equidistant from B and C.

ii. Construct the locus of the vertices of the triangle with BC as base, which are equal in area to triangle ABC.

iii. Mark the point Q in your construction, which would make $Δ$ QBC equal in area to $Δ$ ABC, and isosceles.

iv. Measure and record the length of CQ.

Q. Use ruler and compass only for the following question. All construction lines and arcs must be clearly shown.
Construct a

△ A B C

in which

B C = 6.5 c m, ∠ A B C = 60^{\circ}, A B = 5 c m

. Construct a locus of points

l_{1}

at a distance of

3.5 c m

from

A

. Also, construct a locus of points

l_{2}

which is equidistant from

A C

and

B C

. Let

l_{1}

intersect

l_{2}

in two points

X

and

Y

. Measure

X Y

Use ruler and compass only for this question
(i) Construct $Δ A B C$ , where AB = 3.5 cm, BC = 6 cm and $∠ A B C = 60^{\circ}$
(ii) Construct the locus of points inside the triangle which are equidistant from BA and BC.
(iii) Construct the locus of points inside the triangle which are equidistant from B and C.
(iv) Mark the point P which is equidistant from AB, BC and also equidistant from B and C.

Then the length of PB is

Q. Construct a triangle

A B C

with

A B = 5.5 c m, A C = 6 c m

and

∠ B A X = 105^{o}

Hence :
Construct the locus of points equidistant from

B A

and

B C

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