(English Medium) ICSE Class 10 - CISCE Question Bank Solutions for Mathematics

Describe the locus of vertices of all isosceles triangles having a common base.

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.

Angle ABC = 60° and BA = BC = 8 cm. The mid-points of BA and BC are M and N respectively. Draw and describe the locus of a point which is:

1. equidistant from BA and BC.
2. 4 cm from M.
3. 4 cm from N.
   Mark the point P, which is 4 cm from both M and N, and equidistant from BA and BC. Join MP and NP, and describe the figure BMPN.

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.

Draw an angle ABC = 75°. Find a point P such that P is at a distance of 2 cm from AB and 1.5 cm from BC.

Construct a triangle ABC, with AB = 5.6 cm, AC = BC = 9.2 cm. Find the points equidistant from AB and AC; and also 2 cm from BC. Measure the distance between the two points obtained. 

Construct a triangle ABC, with AB = 6 cm, AC = BC = 9 cm. Find a point 4 cm from A and equidistant from B and C. 

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.

1. Construct a ΔABC, in which BC = 6 cm, AB = 9 cm and angle ABC = 60°.
2. Construct the locus of all points inside triangle ABC, which are equidistant from B and C.
3. Construct the locus of the vertices of the triangles with BC as base and which are equal in area to triangle ABC.
4. Mark the point Q, in your construction, which would make ΔQBC equal in area to ΔABC, and isosceles.
5. Measure and record the length of CQ.

State the locus of a point in a rhombus ABCD, which is equidistant

1. from AB and AD;
2. from the vertices A and C.

Use graph paper for this question. Take 2 cm = 1 unit on both the axes.

1. Plot the points A(1, 1), B(5, 3) and C(2, 7).
2. Construct the locus of points equidistant from A and B.
3. Construct the locus of points equidistant from AB and AC.
4. Locate the point P such that PA = PB and P is equidistant from AB and AC.
5. Measure and record the length PA in cm. 

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. 

Plot the points A(2, 9), B(–1, 3) and C(6, 3) on graph paper. On the same graph paper draw the locus of point A so that the area of ΔABC remains the same as A moves. 

Construct a triangle BCP given BC = 5 cm, BP = 4 cm and ∠PBC = 45°.

1. Complete the rectangle ABCD such that:
   1. P is equidistant from AB and BC.
   2. P is equidistant from C and D.
2. Measure and record the length of AB. 

Prove the following identities:

`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`

Prove the following identities:

`(1 + sin A)/(1 - sin A) = (cosec  A + 1)/(cosec  A - 1)`

Prove the following identities:

`1/(tan A + cot A) = cos A sin A`

Prove the following identities:

`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`

Prove the following identities:

(1 – tan A)² + (1 + tan A)² = 2 sec²A