 # Extra questions for Similar triangles|Class 10 Maths

Question 1.
P and Q are the mid-point of the sides CA and CB respectively of a $\Delta ABC$, right angled at C. Prove that:
(i) $4AQ^2 = 4AC^2 + BC^2$
(ii)$4BP^2 = 4BC^2 + AC^2$
(iii) $4(AQ^2 + BP^2) = 5AB^2$.

Question 2.
The areas of two similar triangles are in the ratio of the squares of the corresponding altitude
Question 3.
The areas of two similar triangles are in the ratio of the squares of the corresponding median.
Question 4.
The areas of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.
Question 5.
If, AD BE and CF are medians of the $\Delta ABC$, then prove that
$3 (AB^2 + BC^2 + CA^2) = 4(AD^2 + BE^2 + CF^2)$

Question 6.
In an equilateral triangle ABC, D is a point on side BC such that $BD = \frac {1}{3} BC$. Prove that $9AD^2 = 7AB^2$

Question 7.
O is a point in the interior of a triangle ABC, $OD \perp BC$, $OE \perp AC$ and $OF \perp AB$. Show that:
a. $OA^2 + OB^2 + OC^2 - OD^2 - OE^2 - OF^2= AF^2 + BD^2 + CE^2$.
b. $AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2$

Question 8.
D is a point on the side BC of a triangle ABC such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB.CD$ Question 9.
D, E and F are respectively the mid-points of sides AB, BC and CA of $\Delta ABC$. Find the ratio of the areas of $\Delta DEF$ and $\Delta ABC$

Question 10.
E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If $AD \perp BC$ and $EF \perp AC$,
prove that $\Delta ABD \sim \Delta ECF$ Question 11.
O is any point inside a rectangle ABCD. Prove that $OB^2 + OD^2= OA^2 + OC^2$.

Question 12.
ABCD is a rectangle. Points M and N are on BD such that $AM \perp BD$ and $CN \perp BD$. Prove that $BM^2 + BN^2 = DM^2 + DN^2$

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### Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20

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