# Worksheet on Theorems of Solid Geometry

Practice the questions given in the worksheet on theorems of solid geometry. Keeping in mind the theorems of solid geometry students need to practice the questions by solving it step-by-step.

1. Find the locus in the three dimensional space of a point equidistant from two given points.

2. Find the locus of a point in space equidistant from three given non-collinear point.

3. O is the circum-centre of a given triangle ABC. If P be any point outside the plane of the triangle ABC such that PA = PB = PC, show that PO is perpendicular to the plane of the triangle ABC.

4. Prove that one and only one perpendicular can be drawn to a plane through a given point outside the plane.

5. The straight line OA, drawn through the centre O of a circle, is perpendicular to the two radii OB and OC of the circle. Prove that all points on the circumference of the circle are equidistant from any points on the line OA.



6. P is a point outside a given plane and O, A, B, C and D are point in the plane such that POA = POB = 1 right angle. If PA = PB = PC = PD, show that the points A, B, C and D are concyclic. Determine the centre of the circle passing through A. B, C and D.

7. How many horizontal lines can be drawn through a given point in a vertical line and how do they lie.

8. If a triangle revolves about its base, prove that its vertex describes a circle. 9. Through the intersection O of the diagonals of a horizontal square ABCD, a vertical line OP is drawn. Prove that, PA = PB = PC = PD.

10. Find a point on a given straight line in space which is equidistant from two given points outside the line. When this impossible?

11. Prove that the straight lines joining the middle points of the opposite sides of a skew quadrilateral bisect each other.

12. The straight lines AB and CD are perpendicular to a plane and meet it at B and D respectively. If the lines are on the same side of the plane and AB = CD, prove that ABCD is a rectangle.

13. P is a point outside the plane of two parallel straight lines AB and CD. From the point P, PL is drawn perpendicular to AB and LM is drawn perpendicular to CD. Show that PM is perpendicular to CD.

14. Two Straight lines AB and AC intersect at right angles. From B a perpendicular BD is drawn to the plane of △ ABC. Prove that AD is perpendicular to the straight line AC.

15. AB, CD, EF are three parallel straight lines not lying in one plane and their extremities form two triangles ACE and BDF. If AB = CD = EF, prove that the triangles are congruent.

Geometry