Question

Given a triangle,If each side of that triangle is doubled, then find the ratio of the new triangle and the initial triangle (use Heron's formula) :)

Answer 1

Answer:

4:1

Step-by-step explanation:

Let a,b,c be the sides of the triangle.

Perimeter 2s = a + b + c

Semi-perimeter, s = (a+b+c)/2

Using Heron's formula:

Area of the triangle A = √s(s−a)(s−b)(s−c)

Now, if the sides are doubled: 2a, 2b, 2c

Let s' be the semi-perimeter.

2s' = 2a + 2b + 2c

s' = a + b + c

or s' = 2s

Area of the triangle, A' = √s′(s′−2a)(s′−2b)(s′−2c)

A' = √(2s)(2s−2a)(2s−2b)(2s−2c)

A' = √24s(s−a)(s−b)(s−c)

A' = 4√s(s−a)(s−b)(s−c)

A' = 4A

A':A = 4:1

Ratio of area of the new triangle and old triangle is 4:1

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Answer 2

Answer:

4:1

Step-by-step explanation:

let one side of a triangle be x

then the corresponding side of this new triangle is 2x because it is doubled

sides of new triangle and given triangle are in proportion

∴given triangle is similar to new triangle

new triangle and the given triangle are similar triangles.

by using area of similar triangles theorem,

area of new triangle/area of given triangle=(2x/x)²

                                                                 =4x²/x²

                                                                 =4/1

∴ratio of area of new triangle and the given triangle=4:1

Hope this helps you!!