Q4 of 13 Page 102

In the picture below, the diagonals of the trapezium ABCD intersect at P.


Prove that PA × PD = PB × PC

Here AB, CD are parallel, then draw a line parallel to AB or CD through P.



Then,


We know that ,


Any three parallel lines will cut any two lines into pieces whose lengths are in the same ratio.


Therefore,


PA×PD = PB×PC


For verification, we can take a small trapezium and prove it as follows:



Here 3.8×2.12 = 7.98 = 1.9×4.25 approximately


More from this chapter

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3

Draw triangles specified below, each of perimeter 10 centimetres.

Sides in the ratio 3 : 4 : 5

 3

Draw triangles specified below, each of perimeter 10 centimetres.

Sides in the ratio 2 : 3 : 4

 1

In the picture, the perpendicular is drawn from the midpoint of the hypotenuse of a right triangle to the base.


Calculate the length of the third side of the large right triangle and the lengths of all three sides of the small right triangle.

 2

Draw a right triangle and the perpendicular from the midpoint of the hypotenuse to the base.


i) Prove that this perpendicular is half the perpendicular side of the large triangle.


ii) Prove that in the large triangle, the distance from the midpoint of the hypotenuse to all the vertices are equal.


iii) Prove that the circumcentre of a right triangle is the midpoint of its hypotenuse.