Relation between Areas and Sides of Similar Triangles

Q.

In a right angled triangle , if a perpendicular is drawn from right angle ti hypotenuse then prove that the area of the square of the perpendicular is equal to the area of the rectangle formed by the two segments of the hypotenuse .

Q.

$\Delta ABC$ is right-angled at A and $AD\perp BC$. If BC = 13 cm and AC = 5 cm, find the ratio of the areas of $\Delta ABCand\Delta ADC$.

Q. Question 7
In the given figure, $\angle Q>\angle R$, PA is the bisector of $\angle QPR$ and PM $\perp$ QR. Prove that $\angle APM=\frac{1}{2}(\angle Q-\angle R)$

Q.

A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire. [1 MARK]

Q. In a ∆ABC, AD is the bisector of ∠BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm. Find AC

(a) 4 cm
(b) 6 cm
(c) 3 cm
(d) 8 cm

Q. In a ∆ABC, AD is the bisector of ∠BAC. If AB = 6 cm, AC = 5 cm and BD = 3 cm, then DC =

(a) 11.3 cm
(b) 2.5 cm
(c) 3 : 5 cm
(d) None of these

Q. In ∆ABC, PQ is a line segment intersecting AB at P and AC at Q such that PQ || BC and PQ divides ∆ABC into two parts equal in area. Find $\frac{\mathrm{BP}}{\mathrm{AB}}$.

Q.

In the adjoining figure, ∆ ABC is right angled at ∠A, CD = 8cm & DB = 18cm. Find AD and the ratio of areas of ∆ ADB : ∆ CDA.

1. 5 cm, 81 : 4

2. 12 cm, 9 : 64

3. 9 cm, 9 : 16

4. 12 cm, 81 : 16

5. 6 cm, 9 : 25

Q.

The corresponding sides of two similar triangles are in the ratio of $2:3$. If the area of the smaller triangle is $48c{m}^2$, find the area of the larger triangle.

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

Q.

In figure, $\frac{AO}{OC}=\frac{BO}{OD}=\frac{1}{2}$ and AB = 5cm. Find the value of DC.

Q.

Base of a triangle is $9$ and height is $5$. Base of another triangle is $10$ and height is $6$. Find the ratio of areas of these triangles.

Q. If ∆ABC ∼ ∆DEF such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of ∆DEF is 25 cm, then the perimeter of ∆ABC is

(a) 36 cm
(b) 30 cm
(c) 34 cm
(d) 35 cm

Q.

In the given figure, DE || BC and DE : BC = 3 : 5. Calculate the ratio of the areas of $\Delta ADE$ and the trapezium BCED.

Q. If two similar triangles have ratio of their areas as 16:25, then the ratio of their perimeters will be

1. 
   9:25
2. 
   3:5
3. 
   4:5
4. 
   16:25

Q.

In the given figure, $\angle ACB={90}^{o}andCD\perp AB$.

Prove that $\frac{B{C}^2}{A{C}^2}=\frac{BD}{AD}$.

Q.

If in a triangle $ABC$, the line joining the circumcentre $O$ and the incentre $I$ is parallel to $BC$, then

1. $r=R\cos A$

2. $r=R\sin A$

3. $R=r\cos A$

4. $R=r\sin A$

Q.

In fig., DE $\parallel$ BC and CD $\parallel$ EF. Prove that AD²= AB x AF.

Q. Two isosceles triangles have equal vertical angles and their areas are in the ratio 36 : 25. Find the ratio of their corresponding heights.

Q. If ∆ABC ∼ ∆DEF such that DE = 3 cm, EF = 2 cm, DF = 2.5 cm, BC = 4 cm, then perimeter of ∆ABC is

(a) 18 cm
(b) 20 cm
(c) 12 cm
(d) 15 cm

Q.

Ina trapezium ABCD, it is given that AB || CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar $\left(\Delta AOB\right)=84c{m}^2$. Find ar $\left(\Delta COD\right)$.

Q. Areas of two similar triangles are 225 sq.cm. 81 sq.cm. If a side of the smaller triangle is 12 cm, then Find corresponding side of the bigger triangle.

Q.

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

Q. In the adjoining figure, DE is parallel to BC and AD = 1 cm, BD = 2 cm. What is the ratio of the area of ∆ABC to the area of ∆ADE?

Q.

Prove that in an equilateral triangle, the circumcentre and incentre are the same. What is the ratio of the circumradius and inradius?

Q.
In the given figure, ∠ACB = 90⁰ and CD ⊥ AB. Prove that

$\frac{{\mathrm{BC}}^2}{{\mathrm{AC}}^2}=\frac{\mathrm{BD}}{\mathrm{AD}}$

Q. Question 136
PQRS is a rectangle. The perpendicular ST from S on PR divides $\angle S$ in the ratio 2 :3. Find $\angle TPQ$.

Q.

If $D,E,F$ are respectively the mid points of the sides $BC,CA$ and $AB$ of $△ABC$ and the area of $△ABC$ is $24$ sq. cm, then the area of $△DEF$ is -

1. $24c{m}^2$

2. $12c{m}^2$

3. $6c{m}^2$

4. $8c{m}^2$

Q.

2 triangles are similar if their corresponding angles are porportional.true or false and justification.

Q.

Two similar triangles have areas $392c{m}^2$ and $200c{m}^2$ respectively. Find the ratio of any pair of corresponding sides.