Scale Factor of Area

Q.

If two similar triangles have a scale factor of a : b, then the ratio of their areas is

1. b : a

2. a${}^3$ : b${}^3$

3. a${}^2$ : b${}^2$

4. 2a : 2b

Q.

$\frac{1}{2}$×base×height is the formula for area of which type of triangles?

1. Only right angled triangles

2. Only isosceles triangles

3. All types of triangles

4. Only acute angled triangles

Q.

Find the ratio of areas of two similar right triangles whose scale factor is 3 : 4.

1. $\frac{16}{9}$

2. $\frac{9}{16}$

3. $\frac{6}{8}$

4. $\frac{27}{64}$

Q. If two similar triangles have a scale factor of a : b, then the ratio of their areas is ______.

1. b : a
2. a^3 : b^3
3. a^2 : b^2
4. 2a : 2b

Q.

The line segment XY is parallel to side AC of Δ ABC and it divides the triangle into two parts of equal areas. Find the ratio $\frac{BX}{AB}$.

1. $\frac{1}{\sqrt{2}}$
2. $\frac{4}{\sqrt{1}}$
3. $\frac{\sqrt{2}}{1}$
4. $\frac{1}{2}$

Q.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
Apply the above theorem to the following :
In a trapezium $ABCD,O$ is the point of intersection of $AC$ and $BD$, $AB\parallel CD$ and $AB=2CD$. If the area of $∆AOB=84{\mathrm{cm}}^2$, find the area of $∆COD$.

Q.

In given figure $△$ ABC and $△$ DEF are similar, BC=4cm, EF=5cm, and area of triangle ABC=32$c{m}^2$ find the area of $△$ DEF

1. $60c{m}^2$

2. $40c{m}^2$

3. $96c{m}^2$

4. $50c{m}^2$

Q.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Apply the above theorem to the following :

$ABC$ is an isosceles triangle right-angled at $B$. Similar triangles $ACD$ and $ABE$ are constructed on sides $AC$ and $AB$ . Find the ratio between the areas of $△ABE$ and $△ACD$.

Q.

If the area of two similar triangles is in the ratio $4:9$, then find the ratio of their corresponding sides

Q.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Apply the above theorem on the following:

$ABC$ is a triangle and $PQ$ is a straight line meeting $AB$ in $P$ and $AC$ in $Q$. If $\mathrm{AP}=1\mathrm{cm},\mathrm{PB}=3\mathrm{cm},\mathrm{AQ}=1.5\mathrm{cm},\mathrm{QC}=4.5\mathrm{cm}$. Prove that the area of $∆APQ$ is one-sixteenth of the area of $∆ABC$.

Q. Area of two congruent triangles ABC and PQR are X and Y respectively.
Which of the following is correct?

1. X > Y
2. X < Y
3. X = Y
4. Comparison cannot be done.

Q.

The base of triangle is $8\mathrm{meters}$. The base of a similar triangle $Q$ is $7\mathrm{meters}$.What is the value of the ratio of the area of $P$ to the area of $Q$?

Q.

In the given, $△ABC~△DEF$. If $AB=2DE$ and the area of $△ABC\text{is}56$ sq.cm, find the area of $△DEF$.

Q. Find the correct pair of ratio of sides and areas of two similar triangles.

1. 49cm² and 81cm²
2. 9cm² and 25cm²
3. 64cm² and 36cm²
4. 8 : 6
5. 3 : 5
6. 7 : 9

Q. Find the ratio of the areas of two similar triangles shown in the figure.

1. ${\left(\frac{AB}{PQ}\right)}^2$
2. ${\left(\frac{BC}{QR}\right)}^2$
3. ${\left(\frac{AC}{PR}\right)}^2$
4. $Allofthese$

Q. Area of two congruent triangles ABC and PQR are X and Y respectively.
Which of the following is correct?

1. X > Y
2. Comparison cannot be done.
3. X < Y
4. X = Y