Application of Similarity

Q.

A storage tank is in the form of a cube. When it is full of water, the volume of water is $15.625m^3$. If the present depth of water is $1.3m,$ then find the volume of water already used from the tank.

Q.

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

1. 15 cm

2. 22.5 cm

3. 7.5 cm

4. 30 cm

Q.

State and prove the converse of the following theorem. “If a line divides any two sides of a triangle in the same ratio it must be parallel to the third side”. Using the result, prove the following:

"A line drawn from the midpoint of a non-parallel side of a trapezium, parallel to the parallel sides, bisects the other non-parallel side.

Q.

Measuring the areas of different currency notes.
Consider the currency notes of denominations 10, 20, 50, 100 and 500 find out the areas of these notes and write in the table given below:

Currency Notes	Length (cm)	Breadth (cm)	Area ( cm2)
10
20
50
100
500

Q.

Question 118

A design is made up of four congruent right triangles as shown in the given figure. Find the area of the shaded portion.

Q. Question 12
Areas of two similar triangles are $36m^2$ and $100c{m}^2$. If the length of a side of the larger triangle is 20 cm. Find the length of the corresponding side of the smaller triangle.

Q.

Question 121

Ramesh grew wheat in a rectangular field that measured 32 metres long and 26 metres wide. This year he increased the area for wheat by increasing the length but not the width. He increased the area of the wheat field by 650 square metres. What is the length of the expanded wheat field?

Q. The perimeter of two similar triangles $ABC$ and $PQR$ are $36cm$ and $24cm$ respectively. If $PQ=10cm$ then the length of $AB$ is

1. $12cm$
2. $18cm$
3. $15cm$
4. $30cm$

Q. Question 54

Fill in the blanks to make the statement true.

1 $k{m}^2$ = ___ $m^2$

Q. Question 4
A rectangular plot is given for constructing a house having a measurement of 40m long and 15m in the front. According to the laws, a minimum of 3m, wide place should be left in the front and back each and 2m wide space on each of other sides. Find the largest area where house can be constructed.

Q.

In the figure given, if $\Delta ABC$ is an isosceles triangle and OB and OC are angle bisectors of $\angle B$ and $\angle C$ respectively, then $\Delta BOC$ is ____.

1. isosceles
2. scalene
3. right-angled

4. equilateral

Q.

Given $△$ DEF $\sim$ $△$ABC. If AB = 3 cm, BC = 2 cm, CA = 2.5 cm and EF = 4 cm, then the perimeter of $△$DEF is ____.

1. 15 cm

2. 20 cm

3. 17 cm

4. 16 cm

Q.

Question 60

State whether the following statement is True or False.

The model of a ship shown is of height 3.5 cm. The actual height of the ship is 210 cm if the scale chosen is 1 : 60.

Q. On a scale map $0.7cm$ represents $8.4km$. If the distance between two points on the map is $46.5cm$, what is the actual distance between the points?

1. $55.80km$
2. $56km$
3. $62.80km$
4. $72km$

Q.

The sides of a triangle are $5,6$ and $10$.

How do you find the length of the longest side of a similar triangle whose shortest side is $15$?

Q.

The angles of a triangle$ABC$ are in A.P. The largest angle is twice the smallest and the median to the largest side divides the angle at the vertex in the ratio $2:3$. If the length of the median is $2\sqrt{3}cm$ then the length of the largest side is

1. $2\sin {32}^{°}$

2. $4\sin {32}^{°}$

3. $6\sin {32}^{°}$

4. $8\sin {32}^{°}$

Q. $△ABC$ and $△PQR$ are similar triangle such that area $\left(△ABC\right)=49{cm}^2$ and Area $\left(△PQR\right)=25{cm}^2$. If $AB=5.6cm$, find the length of $PQ$.

1. $4cm$
2. $5cm$
3. $5.6cm$
4. $7cm$

Q. A man sees the top of a tower in a mirror which is at a distance of $87.6m$ from the tower. The mirror is on the ground, facing upward. The man is $0.4m$ away from the mirror, and the distance of his eye level from the ground is $1.5m$. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line)

Q. In $△ABC,AD$ is the median to $BC$ and in $△PQR$, $PM$ is the median to $QR$. If $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AD}{PM}$. Prove that $△ABC\sim △PQR$.

Q. The map of a country is drawn to a scale of $30cm$ to $50km$. What area does a country, whose area is $4,00,000{km}^2$ cover on a paper?

Q. What is the distance between two cities, $4.5$ cm apart, on a map drawn to scale $1$ cm $=45$ km?

Q.

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

1. $7.5cm$
2. $15cm$
3. $22.5cm$
4. $30cm$

Q. A map is drawn to a scale of $120$ cm to the $1$ km. How many square cm on the map will represent a hectare of ground?

Q. A model of a ship is made to a scale $1:300$. The length of the model of the ship is $2m$. Calculate the length of the ship.

Q. The scale of the map is $20$ km to a cm. What area of a country will be represented on the map, whose area is $60,000$ ${\text{km}}^2$?

Q. Rashmi has a road map with a scale of 1 cm representing 18 km. She drives on a road for 72 km. What would be her distance covered in the map.

Q. The dimensions of the model of a multi-store building are $1.2m\times 75cm\times 2m$. If the scale factor is $1:30;$ then the actual dimensions of the building is $xm\times 22.5m\times 60m$. Find the value of $x$.

Q. In $△ABC$, $AB=3cm,AC=4cm$ and $AD$ is the bisector of $\angle A$. Then $BD:DC$ is:

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

Q.

In the figure given below, sides PB and QA are perpendiculars drawn to the line segment AB.
If PO = 6 cm, QO = 9 cm, and area of $\Delta POB=120c{m}^2$, then find the area of $\Delta QOA$.

Q. A model of a ship is made to a scale $1:300$.
The area of the deck ship is $180,000m^2$. Calculate the area of the deck of the model (in $m^2$).