Problems for Similar triangles|Class 10 Maths

Question 1.
In $\Delta ABC$ , ray AD bisects $ \angle A$ and intersects BC in D. If BC = a, AC = b and AB = c, prove that:
i. $BD = \frac {ac}{b+c}$
ii.$DC = \frac {ab}{c+b}$

Question 2.
Two poles of heights a and b metres are standing vertically on a level ground r metres apart. Prove that the height c of the point of intersection of the lines joining the top of each pole to the foot of the opposite pole is given by $\frac {ab}{a+b}$ i.e $c = \frac {ab}{a+b}$

Question 3.
D and E are points on the sides CA and CB respectively of $\Delta ABC$ right-angled at C. Prove that $AE^2 + BD^2 = AB^2 + DE^2$

Question 4.
In a quadrilateral ABCD, $\angle B = 90$ , $AD^2= AB^2 + BC^2 + CD^2$, prove that $\angle ACD = 90$

Question 5.
ABC is a triangle in which AB = AC and D is a point on AC such that
$BC^2 =AC \times CD$ . Prove that BD=BC

Question 6.
if $ \Delta ABC \sim \Delta DEF$ and their sides are of lengths (in cm) as marked along them, then find the lengths of the sides of each triangle.

Question 7.
True and False statement
(a) All quadrilateral are similar.
(b) All circles are similar.
(c ) All isosceles triangles are similar.
(d) All 30°, 60°, 90° triangles are similar.

Question 8.
In the below figure PA, QB and RC are each perpendicular to AC.
Prove that
$ \frac {1}{x} + \frac {1}{y} = \frac {1}{z}$

Question 9.
ABC is a right triangle right angled at C. Let BC = a, CA = b AB = c and let p be the length of perpendicular from C on AB, prove that
(i) $cp = ab$
(ii) $ \frac {1}{p^2}= \frac {1}{a^2}+ \frac {1}{b^2}$

Question 10.
Diagonals of a trapezium ABCD with AB||DC intersect each other at the point O. If AB = 2DC, find ratio of the areas of AOB and COD

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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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