CBSE Class 10 Mathematics Constructions Notes Set 03

Constructions: Dividing a Segment and Similar Triangles

To divide a segment AB internally in the ratio m : n, draw a ray from A at an acute angle to AB, mark m + n equal steps along it, join the last mark to B and draw a line parallel to this through the m-th mark; it meets AB at the required point. An alternative uses two parallel rays on opposite sides: mark m equal steps from A on one ray and n equal steps from B on the other and join the ends; the join cuts AB in the ratio m : n.
For an external division in the ratio m : n the dividing point lies outside the segment, nearer the end of the smaller part; with AB = d and m not equal to n it is at distance \( \frac{m\,d}{|m-n|} \) from A, on the side away from B if m < n.

Triangle Similar to a Given Triangle

The scale factor is the ratio of the new sides to the corresponding given sides. For a factor \( \frac mn \), draw the triangle, take one side as base, make an acute angle at one end opposite the third vertex, and mark off the larger of m and n as equal steps. Join the n-th point to the other end of the base and draw a parallel through the m-th point; then draw a parallel to the third side. If the factor is less than 1 the new triangle lies inside; if more than 1 the base and the other side are extended and the new triangle is larger.

Tangents to a Circle

At a point P on a circle with centre known, join OP and draw the perpendicular at P. Without the centre, draw a chord PQ and a point R on the circle and make the angle between the tangent and PQ equal to the angle PRQ in the alternate segment. From an external point P, join OP, draw the circle on OP as diameter; its intersections with the given circle are the points of contact because the angle in a semicircle is a right angle. Without the centre: draw a secant PAB, mark C with PC = PA, draw a semicircle on CB and the perpendicular PD to CB; then PD is the tangent length and an arc of radius PD about P cuts the circle at the points of contact.
The tangent length is \( \sqrt{OP^2-r^2} \) and it equals \( \sqrt{PA\cdot PB} \) for any secant PAB.