Relation between Area and Sides of Similar Triangles

Q.

If $△$ABC is similar to $△$DEF such that BC = 3 cm, EF = 4 cm and area of $△$ABC = 54 ${cm}^2$. Determine the area of $△$DEF.

Q. The ratio of sides in two similar triangles is 4:9. What is the ratio of their perimeters?

1. $2:3$
2. $1:2$
3. $3:8$
4. $4:9$

Q.

To the given figure, BC is parallel to DE Area of triangle $ABC=25c{m}^2,$

Area of trapezium BCED $=24c{m}^2$ and DE =14 cm.

The length of BC = ___ cm.

Q.

D, E, F are the mid-points of the sides BC, CA and AB respectively of a $△$ ABC. The ratio of the areas of $△$ ABC and $△$ DEF.

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Q. i) In the figure given below, , $XY\parallel BC$. Given that AX = 3 cm, BX = 1.5 cm and BC = 6 cm. Write down the value of XY

ii) If $\frac{AX}{XB}=\frac{4}{13}$ and AC = 20.4 cm, find AY. [3 MARKS]

Q. Question 172
ABCD is a parallelogram. Find the values of x, y and z.

Q.

In the given figure, AX : XB = 3 : 5 and XY || BC.

Then, the ratio of the areas of trapezium XBCY and triangle ABC is

1. 64:45

2. 45:64

3. 64:55

4. 55:64

Q.

In a $△ABC$, BD and CE are the altitudes. Prove that $△ADBand△AEC$ are similar. Is $△CDB\sim △BEC$ ?

Q.

If ABC is an equilateral triangle of side a, prove that its altitude = $\frac{\sqrt{3}}{2}$a

Q.

In the given figure, $\angle ABC=\angle AED={90}^{\circ }$, AB = 12 cm, AC = 15 cm and DE = 3 cm. The area of $\Delta ABC$ and $\Delta AED$ are ___ and ___ respectively.

1. $54c{m}^2,6c{m}^2$

2. $52c{m}^2,8c{m}^2$

3. $56c{m}^2,6c{m}^2$

4. $52c{m}^2,6c{m}^2$

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. 0.525 cm

4. 4.2 cm

Q. Question 4
Diagonals of a trapezium PQRS intersect each other at the point O, PQIIRS and PQ = 3 RS. Find the ratio of the areas of $\Delta$POQ and $\Delta$ROS.

Q. In the given figure, AB and DE are perpendicular to BC. [3 MARKS]

i) Prove that $\Delta ABC\sim \Delta DEC$
ii) If AB = 6 cm, DE = 4 cm and AC = 15 cm, calculate CD.
iii) Find the ratio of the area of $\Delta ABC$ : area of $\Delta DEC$

Q.

In $\Delta$ABC, if a triangle is formed by joining the midpoints of the sides, the area of the triangle is___ times the area of $\Delta$ ABC.

1. $\frac{1}{3}$

2. 4

3. 2

4. $\frac{1}{4}$

Q.

The two figures are similar.

1. Find the value of $x$.
2. Find the values of the ratios (red to blue) of the perimeters and of the areas.

Q.

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

Q. Question 5
D, E and F are respectively the mid-points of sides AB, BC and CA of $\Delta ABC$. Find the ratio of the area of $\Delta DEF$ and $\Delta ABC$.

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.

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Q.

If $△$ABC ~ $△$DEF such that AB = 1.2 cm and DE = 1.4 cm . Find the ratio of area of $△$ ABC and $△$DEF.

Q.

The triangles $ABC$ and $DFG$ are similar and the ratio of their corresponding sides is $6:5$.

The area of the triangle $ABC$ is greater than the area of the triangle $DFG$ by $77{\mathrm{cm}}^2$.

Find the areas of these triangles.

Q.

In the given figure, ABC is a triangle. DE is parallel to and $\frac{AD}{DB}=\frac{3}{2}$

(i) Determine the ratio $\frac{AD}{AB}$.

(ii) What is the ratio of the areas of $\Delta DEF$ and $\Delta BFC?$

1. (i) $\frac{3}{5}$, (ii) 9 : 25

2. (i) $\frac{2}{5}$, (ii) 9 : 25

3. (i) $\frac{2}{3}$, (ii) 4 : 9

4. (i) $\frac{1}{2}$, (ii) 4 : 9

Q. (a) If the areas of two similar triangles are equal, prove that they are congruent.

(b) The scale of the map is 1 : 200000. A plot of land with area 20 $k{m}^2$ is to be represented on the map. Find:

(i) The number of kilometres on the ground which is represented by 1 cm on the map.
(ii) The area in sq. km that can be represented by $1c{m}^2$
(iii) The area of the map that represents the plot of land. [6 MARKS]

Q. In the given figure, ABC is a triangle with $\angle EDB=\angle ACB$. Prove that $\Delta ABC\sim \Delta EBD.IfBE=6cm,EC=4CM,bd=5cmandareaof\Delta BED=9c{m}^2$, calculate: [4 MARKS]
(i) Leight of AB
(ii) Area of $\Delta ABC$.

Q.

Given that BC and BC' lie on the same ray BY. The ratio of $\frac{BC}{C{C}^{\prime }}$ is 3:5.
What will the ratio of corresponding sides be if triangles ABC and A'BC' are similar?

1. 5:8

2. 3:8

3. 8:5

4. 8:3

Q.

In $\Delta$ABC, if a triangle is formed by joining the midpoints of the sides, the area of the triangle is___ times the area of $\Delta$ ABC.

1. 2

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

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

4. 4

Q. Question 11

Ratio of areas of $\Delta MNO,\Delta MOPand\Delta MPQ$ in the given figure is



(a) 2 : 1 : 3
(b) 1 : 3 : 2
(c) 2 : 3 : 1
(d) 1 : 2 : 3

Q. Areas of two similar triangles are 49 $c{m}^2$ and 100 $c{m}^2$. If the length of a side of the smaller triangle is 14 cm, the length of the corresponding side of the other triangle is:

1. 17 cm
2. 19 cm
3. 18 cm
4. 20 cm

Q. In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Q. Question 4 (i)
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
(i)$\Delta ABE\cong \Delta ACF$

Q.

D, E, F are the mid-points of the sides BC, CA and AB respectively of a $\Delta$. Then the ratio of the areas of $\Delta DEF$ and $\Delta ABC$ is

1. 4 : 1

2. 1 : 2

3. 2 : 1

4. 1 : 4