**NCERT Class 10 Maths Lab Manual – Ratio of Areas of Two Similar Triangles**

**Objective**

To verify **“The ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides”** by performing an activity.

**Prerequisite Knowledge**

- Concept of parallel lines.
- Division of a line in a given ratio.

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**Materials Required**

Chart paper, construction box, coloured pens, a pair of scissors, fevicol.

**Procedure**

- Take a chart paper and cut a ∆ABC with AB = 6cm, BC = 6cm, CA = 6cm.
- Mark 5 points P
_{1}, P_{2}, …. , P_{5}at a distance of 1cm each on side AB and Q_{1}, Q_{2},…..,Q_{5}at a distance of 1cm each on side AC as shown in fig.(i).

- Join P
_{1}Q_{1}, P_{2}Q_{2}, …………, P_{5}Q_{5}as shown in fig. (ii).

- Draw lines parallel to AC from P
_{1}, P_{2}, P_{5}and also draw lines parallel to AB from the points Q_{1}, Q_{2}, ……., Q_{5}as shown in fig. (iii).

- Thus ∆ABC is divided into 36 smaller triangles and all are similar to each other and of equal area.
- Construct a ∆PQR with PQ = \(\frac { 1 }{ 2 }\) of AB, PR = \(\frac { 1 }{ 2 }\) of AC and QR = \(\frac { 1 }{ 2 }\) of BC i.e. 3cm each on another chart paper.
- Mark D
_{1}, D_{2}and E_{1}, E_{2}on sides PQ and PR respectively. - Repeat steps 3 and 4.
- Thus ∆PQR is divided into 9 smaller similar triangles equal in area.

**Observation**

- area of ∆ABC = area of 36 smaller ∆’s
- area of ∆PQR = area of 9 smaller ∆’s
- \(\frac {PQ}{AB}=\frac {3}{6} =\frac {1}{2} =\frac {PR}{AC}\)
- \(\frac { Area\quad of\quad \Delta PQR }{ Area\quad of\quad \Delta ABC } =\frac { { PQ }^{ 2 } }{ { AB }^{ 2 } }\)

= [9 smaller Δ’s/36 smaller Δ’s = \(\frac { 1 }{ 4 }\) = (1/2)^{2}

(because ΔABC ∼ ΔPQR)

**Result**

Thus it is verified that the ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

**Learning Outcome**

Concept of area theorem is clear to the students through this activity.

**Activity Time**

1. Take isosceles similar triangles and scalene similar triangles and try to verify this activity. Here isosceles triangles, ∆ABC ~ ∆PQR. Scalene triangle ∆DEF ~ ∆KLM

**Viva Voce**

**Question 1.**

What are the criteria for two triangles to be similar ?

**Answer:**

Two triangles are said to be similar, if

- their corresponding angles are equal.
- their corresponding sides are in proportion

**Question 2.**

∆ABC ~ ∆DEF and their areas are respectively 64 cm^{2} and 121 cm^{2}. If EF = 15.4 cm, then findBC.

**Answer:**

11.2 cm

**Question 3.**

Is it true, if the areas of two similar triangles are equal, then they are congruent ?

**Answer:**

Yes

**Question 4.**

What is the ratio of the area of an equilateral triangle described on one side of a square to the area of an equilateral triangle described on one of its diagonal ?

**Answer:**

1:2

**Question 5.**

Are a square and a rhombus of side 3 cm similar ?

**Answer:**

No

**Question 6.**

Is a rhombus of side 3 cm congruent to another rhombus of side 4 cm ?

**Answer:**

No

**Question 7.**

Is the ratio of the areas of two similar triangles equal to the square of the ratio of their corresponding medians ?

**Answer:**

Yes

**Multiple Choice Questions**

**Question 1.**

ABC and BDE are two equilateral triangles, such that D is the mid-point of BC. The ratio of the areas of ΔABC and ΔBDE is

(a) 2:1

(b) 1:2

(c) 4:1

(d) 1:4

**Question 2.**

Sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

(a) 2:3

(b) 4:9

(c) 81:16

(d) 16:81

**Question 3.**

If in two similar triangles PQR and LMN, if QR =15 cm and MN = 10 cm, then the ratio of the areas of triangles is

(a) 3:2

(b) 9:4

(c) 5:4

(d) 7:4

**Question 4.**

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

(a) 16 : 25

(b) 256 : 625

(c) 4 : 5

(d) none of these

**Question 5.**

∆ABC ~ ∆DEF. If AC = 19 cm and DF = 8 cm, then the ratio of the areas of the two triangles is

(a) 361 : 64

(b) 19 : 8

(c) 19 : 4

(d) none of these

**Question 6.**

In the given figure, PB and QA are perpendicular to segment AB. If PO = 5 cm, QO = 7 cm and area (∆POB) = 150 cm^{2}, then area of ∆QOA is

(a) 254 cm^{2}

(b) 294 cm^{2}

(c) 244 cm^{2}

(d) 49 cm^{2}

**Question 7.**

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.

(a) 4:1

(b) 1:4

(c) 4:4

(d) 2:4

**Question 8.**

ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, then the ratio of area (∆ABC) to area (∆DBC) is

(a) \(\frac {AO}{DO}\)

(b) \(\frac {AO}{ DB}\)

(c) \(\frac {AC}{DO}\)

(d) None of these

**Question 9.**

Area (∆ABC) : Area (∆DEF) = 25 : 36. Then AB : DE is

(a) 625 : 1296

(b) 25 : 36

(c) 6 : 5

(d) 5 : 6

**Question 10.**

∆DEF ~ ∆ABC; If DE : AB = 2 : 3 and area ∆DEF is equal to 44 square units, then area (∆ABC) is

(a) 120 sq. units

(b) 99 sq. units

(c) 66 sq. units

(d) none of these

**Answers**

- (c)
- (d)
- (b)
- (c)
- (a)
- (b)
- (a)
- (a)
- (d)
- (b)

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