10th - Similar Triangles IV

Class-X

Triangles-2

1. P and Q are points on sides AB and AC respectively of ∆ABC. If  P=3cm,PB= 6 cm,AQ=5 cm and QC=10 cm, show that BC=3PQ
2.  A man goes 15 m due west and then 8 m due north.Find his distance    from the starting point.
3.  D,E,F are the mid-points of the sides BC,CA and AB respectively of  ∆ABC.Determine  the ratio of the areas of triangle DEF and triangle ABC 
4.  ABCD is a trapezium with AB║DC.If ∆ AED is similar to ∆BEC,prove  that AD=BC.
5. In a right angled  triangle with  side a and b and hypotenuse c, the   altitude drawn on the hypotenuse is x. Prove that: ab=cx.
6. Find the area of ∆ABC in which ےABC=90,ےACB=45 and AC=8 cm.
7. The diagonal BD of a parallelogram ABCD intersects AE at a point F where E is any point on the side BC.Prove that: DF.EF= FB.FA.
8. Two triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and DV intersect at P.Prove that:  AP x PC= BP x PD. 
9. Prove that the area of the triangle BCE described on one side BC of a square ABCD as base is one- half the area of the similar triangle ACF described on the diagonal AC as  base.
10. Find the area of a right triangle, the radius of whose circumcircle is 3 cm and the length of the altitude drawn from the opposite vertex to the hypotenuse is 2 cm. 
11. A right triangle has hypotenuse of length q cm and one side p cm.. If(q-p)=2,express the length of third side of the  right triangle in terms of q.