Q3 of 13 Page 106

In the parallelogram ABCD, the line drawn through a point P on AB, parallel to BC, meets AC at Q. The line through Q, parallel to AB meets AD at R.


i) Prove that


ii) Prove that

i) Here we have AD, PQ, BC as three parallel lines.



We know that


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


For triangle ABC,


Therefore, --------1


Similarly, for triangle ACD,


--------2


From 1 & 2,



Hence proved


ii) From previous part we have,



On inverting the numerators and denominators,



On adding one on both sides of equation,



From figure, on simplifying,



On inverting the numerators and denominators,



Hence proved.


More from this chapter

All 13 →
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.

 4

In the picture below, two vertices of a parallelogram are joined to the midpoints of two sides.


Prove that these line divide the diagonal in the picture into three equal parts.

 5

Prove that the quadrilateral formed by joining the midpoints of a quadrilateral is a parallelogram. What if the original quadrilateral is a rectangle? What if it is a square?