Ratio of Area of Similar Triangles

Q.

P and Q are points on the sides AB and AC respectively of $△$ABC such that PQ||BC and divides $△$ABC into two parts, equal in area. Find PB : AB.

Q. The ratio of sides of 2 similar triangles is 3:4. Find the ratio of their areas.

1. $3:4$
2. $4:3$
3. $9:16$
4. $16:9$

Q. In $\Delta DEW,AB||EW$. If AD = 4 cm, DE = 12 cm and DW = 24 cm, then find the value of DB.

Q.

In the given figure, $△ABC$ and $△DEF$ are similar. If BC = 3 cm, EF = 4 cm and area of $△ABC$ is 54 cm${}^2$, then the area of $△DEF$ is _____ cm${}^2$.

1. 96
2. 48
3. 24
4. 64

Q.

The areas of two similar triangles are $9c{m}^{2}and16c{m}^2$ respectively . The ratio of their corresponding sides is ____.

1. 3: 4

2. 4: 3

3. 4: 5

4. 2: 3

Q.

Two isosceles triangles have equal angles and their areas are in the ratio 16 : 25. The ratio of corresponding heights is :

1. 5: 4

2. 3: 2

3. 5: 7

4. 4: 5

Q.

The areas of two similar triangles are $12c{m}^2$ and $48c{m}^2$. If the height of the smaller one is 2.1 cm, then the corresponding height of the larger one is

1. 4.2 cm

2. 4.41 cm

3. 8.4 cm

4. 0.525 cm

Q. In ΔABC, D and E are the mid- points of AB and AC respectively. Find the ratio of the areas ΔADE and ΔABC.

1. 1 : 5
2. 2 : 5
3. 2 : 3
4. 1 : 4

Q.

In an equilateral triangle, prove that the centroid and the centre of the circumcircle (circumcentre) coincide. [4 MARKS]

Q.

$ABCD$ is a square. Equilateral triangles $ACF$ and $ABE$ are drawn on the the diagonal $AC$ and side $AB$ respectively. Find area of $△ACF$ : area of $△ABE$.

1. $4:1$

2. $\sqrt{2}:1$

3. $8:1$

4. $2:1$

Q.

O is any point inside a triangle ABC. The bisector of $\angle$AOB, $\angle$BOC and $\angle$COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD $\times$ BE $\times$ CF = DB $\times$ EC $\times$ FA. [4 MARKS]

Q.

In the given figure, ABC is a triangle. DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$. If the area of $\Delta DEF$ is $81c{m}^2$. The area of $\Delta CFB$ is

1. $180c{m}^2$

2. $250c{m}^2$

3. $225c{m}^2$

4. $195c{m}^2$

Q. In the given figure, if $PM=10cm,$ $A\left(△PQS\right)=100sqcm,$ and $A\left(△QRS\right)=110sqcm$, then find the measure of $NR$.
[2 Marks]

Q.

AM and PN are the medians of two similar triangles, $△$ABC and $△$PQR respectively and $\frac{Areaof△ABC}{Areaof△PQR}$ = $\frac{9}{25}$.

If AM = PO = 5 cm, find the value of $3ON$.

1. 10 cm
2. 12 cm
3. 13 cm
4. 9 cm

Q. P and Q are points on the sides AB and AC respectively of $△$ABC such that PQ || BC and divides $△$ABC into two parts, equal in area. Find PB : AB.

1. $\frac{1}{\sqrt{2}}$
2. $\frac{(\sqrt{3}-1)}{\sqrt{3}}$
3. $\frac{(\sqrt{2}-1)}{\sqrt{2}}$
4. $\frac{(\sqrt{3}-1)}{\sqrt{2}}$

Q. In $\Delta ABC$ and $\Delta APQ$, AP = 5 cm and PB = 10 cm. Find the ratio of the areas of $\Delta ABCand\Delta APQ.$

1. 1 : 9
2. 3 : 1
3. 1 : 3
4. 9 : 1

Q.

Question 3
Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.

Q. In the adjacent figure, P and Q are points on the sides AB and AC respectively of a triangle ABC. PQ is parallel to BC and divides the triangle ABC into 2 parts, equal in area. The ratio of $PB:AB=$

1. $1:1$
2. $(\sqrt{2}-1):1$
3. $(\sqrt{2}-1):\sqrt{2}$
4. $1:\sqrt{2}$

Q. ABC is a triangle in which ∠ A = 90°, AN ⊥ BC, BC = 12 cm and AC = 5 cm. Find the ratio of the areas of ΔANC and ΔABC.

1. 13 : 45
2. 14 : 93
3. 25 : 144
4. 1 : 5

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 $c{m}^2$, then find the area of $△$ DEF.

1. 12 $c{m}^2$

2. 8 $c{m}^2$

3. 24 $c{m}^2$

4. 6 $c{m}^2$

Q.

In given figure $△$ ABC and $△$ DEF are similar, BC=3cm, EF=4cm, and area of triangle ABC=54$c{m}^2$ , then the area of $△$ DEF

1. 96cm2
2. 96$m^2$
3. 9.6$m^2$
4. 96$c{m}^2$

Q.

$△ABC$ is such that $AB=3cm,BC=2cm$ and $CA=2.5cm.$ $△DEF$ is similar to $△ABC.$ If $EF=4cm$, then the perimeter of $△DEF$ is ____.

1. 7.5 cm

2. 15 cm

3. 22.5 cm

4. 30 cm

Q. In the figure, DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$. Area of $\Delta DEF$ is $81c{m}^2$. Area of (\Delta CFB\) is .

1. $180c{m}^2$
2. $250c{m}^2$
3. $225c{m}^2$
4. $195c{m}^2$

Q. Question 2 (ii)
E and F are points on the sides PQ and PR respectively of a $\Delta PQR$. For the following case, state whether EF || QR.
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

Q. In the figure, DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$. Area of $\Delta DEF$ is $81c{m}^2$. Area of $\Delta CFB$ is .

1. $180c{m}^2$
2. $250c{m}^2$
3. $225c{m}^2$
4. $195c{m}^2$

Q.

The areas of two similar triangles are 12 $c{m}^2$ and 48 $c{m}^2$. If the height of the smaller triangle is 2.1 $cm$, then the corresponding height of the bigger triangle is _____.

1. 4.41 cm

2. 8.4 cm

3. 4.2 cm

4. 0.525 cm

Q.

In given figure $△$ ABC and $△$ DEF are similar, EF=3cm, BC=4cm, and area of triangle DEF=54$c{m}^2$ . Find the area of $△$ ABC.

___ $c{m}^2$.

Q. Question 8
In figure, if DEIIBC, find the ratio of ar $\left(\Delta ADE\right)$ and ar $\left(\Delta DECB\right)$.

Q.

$△$ABC and $△$PQR are two similar triangles as shown in the figure such that $\frac{AreaofABC}{AreaofPQR}$ = $\frac{9}{25}$. AM and PN are the medians on $△$ABC and $△$PQR respectively. If AM = PO = 5 cm, find the value of 3ON.

__

Q. In the figure, DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$. Area of $\Delta DEF$ is $81c{m}^2$. Area of $\Delta CFB$ is .

1. $180c{m}^2$
2. $250c{m}^2$
3. $225c{m}^2$
4. $195c{m}^2$