Construction

Exercise

1. Draw a circle of radius 3 cm. Mark its center as O. Mark a point P such that OP = 5 cm. Using ruler and compasses only, construct the two tangents from P to the circle.
2. Draw a circle of radius 2·5 cm. Construct two tangents to it inclined at an angle of 60° to each other.
3. Construct a circle of radius 2 cm to touch a given circle of radius 3 cm externally. Construct a direct common tangent to these two circles.
4. Draw an equilateral triangle of side 3·5 cm. Construct its circumcircle.
5. Construct a triangle ABC with BC = 4·5 cm, AB = 3·8 cm and C = 75°. Draw circumcircle of ABC.
6. Construct a triangle ABC, given that BC = 5 cm, B = 67½° and CA = 4·5 cm. Construct the inscribed circle of ABC.
7. Draw a straight line ABC such that AB = 4 cm and BC = 3 cm. Using ruler and compasses only, find a point P equidistant from B and C, such that APC = 90°.
   [Hint. Draw a circle with AC as diameter. Draw perpendicular bisector of BC to meet the circle in point P.]
8. Using ruler and compasses only, construct ABC given that BC = 6 cm, B = 60° and AB = 5 cm. Draw the bisector of A to meet BC at D. Find, by construction, the point P which is equidistant from A, C and D.
9. Construct an equilateral triangle in a circle of radius 2·4 cm.
10. Construct an equilateral triangle about a circle of radius 1·8 cm.
11. Draw a circle of radius 2·8 cm and inscribe a square in it.
12. Draw a circle of radius 2·6 cm. Using ruler and compasses only, construct a regular figure of four sides about it.
13. Draw two circles of radii 4·5 cm and 2·5 cm respectively such that their centers are 8 cm apart. Using ruler and compasses only, construct a direct common tangent to these circles.
14. Draw two circles of radii 4 cm and 2·5 cm respectively such that their centers are 9 cm apart. Using ruler and compasses only, construct a transverse common tangent to these circles.
15. Construct a circle of radius 2 cm to touch a given circle of radius 3·5 cm externally. Also construct all direct common tangents to these two circles.
16. The center O of a circle of radius 2 cm is at a distance 5 cm from a given straight line AB. P is a point on AB such that OP = 6 cm. Draw a circle to touch the line AB at P and to touch the given circle externally.
17. Draw a circle of radius 2·2 cm with center O. Take a point P such that OP = 4·5 cm. Construct a tangent PT from P to the circle. Locate a point Q on the circle such that PQT = 90°. Construct a circle to touch the circle at Q and to pass through P.
18. Draw a circle of radius 3 cm. At a point P on the circle, construct the tangent PT. Take a point Q on the tangent PT so that PQ = 3·6 cm. Construct the circle to touch the circle externally and to touch the line PT at Q.
19. Construct a ABC given that AB = 5 cm, BC = 4·5 cm and A = 75°. Construct a circle of radius 2·4 cm to touch AB and AC.
20. Using ruler and compasses only, draw a sector of a circle of radius 5 cm and angle at the center = 60°. Construct another circle to touch the two radii and the arc of the circle (i) internally (ii) externally.
21. Ruler and compasses only 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 triangle ABC, in which AB = 9 cm, BC = 10 cm and angle ABC = 45°.
    (ii) Draw a circle, with center A and radius 2·5 cm. Let it meet AB at D.
    (iii) Construct a circle to touch the circle with center A externally at D and also to touch the line BC.
22. Draw line segment OA, 5 cm long. At O, using ruler and compasses only, construct OB such that AOB = 67½°. Construct a circle to touch OA at A and to touch OB.
23. Construct a ABC given that BC = 4 cm, C = 75° and the radius of the circumcircle of ABC to be 3cm. Construct the circle touching BC at its mid-point and also touching side AC of ABC.
24. Construct a ABC in which AB = 5 cm, BC = 6 cm and CA = 4 cm. Construct a circle passing through B and C and which has AB as a tangent.
25. On a line segment AB = 6 cm, draw a segment of a circle containing an angle of 60°. Find, by construction, a point C such that area of ABC = 12 cm² and ACB = 60°.
26. Using ruler and compasses only, construct a right angled triangle ABC such that hypotenuse BC = 10 cm, the altitude to the hypotenuse is 4·8 cm and A is nearer to B than C. Construct a chord CD of length 6 cm. Hence prove that AD is parallel to BC.
27. Construct a ABC, having given that the radius of the circumcircle of ABC is 3·5 cm, C = 45° and A = 60°.
28. Construct a ABC in which BC = 4·4 cm, A = 60° and the altitude through A = 3·6 cm.
29. Construct a ABC in which BC = 6 cm, A = 75° and median through A is 3·5 cm long.