# Class 10 Maths Proof questions for Similar triangles

1) P and Q are the mid – point of the sides CA and CB respectively of a  ABC, right angled at C. Prove that:
(i)     4AQ2 = 4AC2 + BC2
(ii)     4BP2 = 4BC2 + AC2
(iii)    4(AQ2 + BP2) = 5AB2.
2) The areas of two similar triangles are in the ratio of the squares of the corresponding altitude
3) The areas of two similar triangles are in the ratio of the squares of the corresponding median.
4) The areas of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.
5) If, AD BE and CF are medians of the  ABC, then prove that
3 (AB2 + BC2 + CA2) = 4(AD2 + BE2 + CF2).
6) In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3 BC. Prove that 9AD2 = 7AB2
7) O is a point in the interior of a triangle ABC, OD - BC, OE - AC and OF - AB. Show that:
a) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2.
b)   AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
8) D is a point on the side BC of a triangle ABC such that ∠ ADC = ∠ BAC. Show that CA2 = CB.CD

9) D, E and F are respectively the mid – points of sides AB, BC and CA of  ABC. Find the ratio of the areas of  DEF and  ABC
10)  E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC,
prove that Δ ABD ~ Δ ECF

11) O is any point inside a rectangle ABCD. Prove that OB2 + OD2 = OA2 + OC2.
12) ABCD is a rectangle. Points M and N are on BD such that AM - BD and CN - BD. Prove that BM2 + BN2 = DM2 + DN2