Question

Two triangles are similar; if their corresponding angles are equal and their corresponding
sides are in the same ratio (or proportional).Also the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

Now answer the following Questions-

1- If the sides of a triangle are increased by 50% to make a similar triangle. What is the percentage increase in area?
A) 50%
B) 100 %
C) 125 %
D) 75%

2- what will be the ratio of areas of original triangles to the similar triangle made In the above question?
A)1
B)4
C)3
D)16

3- by what percent the area will decrease if sides of triangle is decreased by 50 percent ?
A)50٪
B)100%
C)125%
D)75%​

Answer 1

Given:

Two triangles are similar; if their corresponding angles are equal and their corresponding  sides are in the same ratio (or proportional).

Also, the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

To find:

(1). If the sides of a triangle are increased by 50% to make a similar triangle. What is the percentage increase in the area?

(2). What will be the ratio of areas of original triangles to the similar triangle made In the above question?

(3) By what percent the area will decrease if sides of the triangle are decreased by 50%?

Solution:

Case (1): Finding the % increase in the area:

Let's assume,

"a" → the side of the original triangle

"A₁" → the area of the original triangle

"A₂" → the area of the new triangle with side increased by 50%.

We have a similar triangle with sides increased by 50%

So, The side of the new triangle = $a + \frac{50}{100}a = \frac{150a}{100} = 1.5a$

Since A₁ ~ A₂

∴ $\frac{A_1}{A_2} = (Ratio\:of\:the\:squares\:of\:any\:two\:corresponding\:sides)$

$\implies \frac{A_1}{A_2} = \frac{a^2}{(1.5a)^2}$

$\implies \frac{A_1}{A_2} = \frac{a^2}{2.25 a^2}$

$\implies \frac{A_1}{A_2} = \frac{1}{2.25 }$

$\implies A_2 =2.25 A_1$

Thus,

The percentage increase in the area is,

= $\frac{A_2 - A_1}{A_1} \times 100$

= $\frac{2.25A_1 - A_1}{A_1} \times 100$

= $\frac{1.25A_1 }{A_1} \times 100$

= $\underline{\bold{125\%}}$

Case (2): Finding the ratio of areas:

Let's assume,

"A₁" → the area of the original triangle

"A₂" → the area of the new triangle with side increased by 50%.

We have a similar triangle with sides increased by 50%

So, The side of the new triangle = $a + \frac{50}{100}a = \frac{150a}{100} = 1.5a$

Since A₁ ~ A₂

∴ $\frac{A_1}{A_2} = (Ratio\:of\:the\:squares\:of\:any\:two\:corresponding\:sides)$

$\implies \frac{A_1}{A_2} = \frac{a^2}{(1.5a)^2}$

$\implies \frac{A_1}{A_2} = \frac{a^2}{2.25 a^2}$

$\implies \frac{A_1}{A_2} = \frac{1}{2.25 }$

$\implies \frac{A_1}{A_2} = \frac{100}{225 }$

$\implies \underline{\bold{\frac{A_1}{A_2} = \frac{4}{9 }}}$

Case (3): Finding the % decrease in the area:

Let's assume,

"a" → the side of the original triangle

"A₁" → the area of the original triangle

"A₂" → the area of the new triangle with side decreased by 50%.

We have a similar triangle with sides decreased by 50%

So, The side of the new triangle = $a - \frac{50}{100}a = \frac{50a}{100} = 0.5a$

Since A₁ ~ A₂

∴ $\frac{A_1}{A_2} = (Ratio\:of\:the\:squares\:of\:any\:two\:corresponding\:sides)$

$\implies \frac{A_1}{A_2} = \frac{a^2}{(0.5a)^2}$

$\implies \frac{A_1}{A_2} = \frac{a^2}{0.25 a^2}$

$\implies \frac{A_1}{A_2} = \frac{1}{0.25 }$

$\implies A_2 =0.25 A_1$

Thus,

The percentage increase in the area is,

= $\frac{A_1 - A_2}{A_1} \times 100$

= $\frac{A_1 - 0.25A_1}{A_1} \times 100$

= $\frac{0.75A_1 }{A_1} \times 100$

= $\underline{\bold{75\%}} }$

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