In the given fig., a circle is inscribed in a triangle abc, such that it touches the sides ab, bc and ca at points d, e and f respectively. if the lengths of sides ab, bc and ca are 12 cm, 8 cm and 10 cm respectively, find the lengths of ad, be and cf.

Find the ratio in which y-axis divides the line segment joining the points a(5, – 6) and b(– 1, – 4). also find the coordinates of the point of division.

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In fig., a quadrilateral abcd is drawn to circumscribe a circle, with centre o, in such a way that the sides ab, bc, cd and da touch the circle at the points p, q, r and s respectively.
prove that: ab+ cd = bc + da.

Let p and q be the points of trisection of the line segment joining the points a(2, – 2) and b(– 7, 4) such that p is nearer to a. find the coordinates of p and q.

Given the linear equation x – 2y – 6 = 0, write another linear equation in these two variables, such that the geometrical representation of the pair so formed is :
i) coincident lines (ii) intersection lines

In the given figure, pa and pb are tangents to the circle from an external point p. cd is another tangent touching the circle at q. if pa = 12 cm, qc= qd = 3 cm, then find pc + pd.

25. prove that tangents drawn at the ends of a diameter of a circle are parallel to each other.
26. find the value of k for which the equation x2 + k(2x+ k – 1) + 2 = 0 has real and equal roots.