Q2 of 70 Page 139

In □m ABCD, T ∈ and intersects in M and in O. Prove that AM2 = MT • MO.

In ABCD, T and intersects in M and in O.



To prove:


AM2 = MT •MO


In ∆AMB and ∆OMD


∠AMD ≅ ∠OMD (Vertically opposite angles)


∠MBA ≅ ∠MDO (Alternate angles)


The correspondence AMD↔OMB is a similarity


…(1)


In ∆AMD and ∆TMB


∠AMD ≅ ∠TMB (Vertically opposite angles)


∠MAD ≅ ∠MTB (Alternate angles)


The correspondence AMD↔TMB is a similarity


…(2)


Multiplying (1) and (2),




Hence, AM2 = MT •MO


More from this chapter

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7

State giving reasons, whether the following statements are true or false: In all the following questions the line does not contain a side of the triangle.

(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.


(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.


(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.


(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.


(5) In the plane of Δ ABC, a line / can be drawn such that = {P}, = {Q} and = ϕ.

 1

∠B is a right angle in ΔABC and is an altitude to hypotenuse. AB = 8, BC = 6. Find the area of ΔBDC.

 3

In □m ABCD, M is the mid-point of . and intersect in N. Prove that DN = 2MN.

 4

P and Q are the mid-points of and in ΔABC. If the area of ΔAPQ = 12√3, find the area of ΔABC.