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

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

Prove the following identities:

cosec⁴ A – cosec² A = cot⁴ A + cot² A

Prove the following identities:

cosec A(1 + cos A) (cosec A – cot A) = 1

Prove the following identities:

sec²A + cosec²A = sec²A . cosec²A

Prove the following identities:

`((1 + tan^2A)cotA)/(cosec^2A) = tan A`

Prove the following identities:

cot² A – cos² A = cos² A . cot² A