Question

when you dilate a triangle with vertices (1,2), (2,4) and (3,-1) with the origin as the centre of dilation, what would the vertices of the new triangle be if the scale factor was 2?



answer immediately . it is from transformation or trigonometry
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Answer 1

Answer:

Dilation by scale factor of 2 means that the dimensions {here all sides of the triangle} have been doubled.

The centre will remain the same. and the sides of new triangle will be parallel to the sides of the initial triangle.

Area of new triangle will be quadrupled. as sides have doubled.

we get this from similarity of the two triangles

Answer:

(4,6), (8,-8), (4,-4)

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Solution :

let one of the vertices of the new triangle be (x₁ , y₁)

and other be (x₂ , y₂).

let's take

(x₁ , y₁) to be the corresponding point to (2,3)

and (x₂ , y₂) to be of (4,-4)

$\sqrt{ {(2 - 0)}^{2} + {(3 - 0)}^{2} }$

$\sqrt{4 + 9}$

$\sqrt{13}$

so the distance of (x₁ , y₁) from origin will be double of it i.e 2√13 .

below x and y represents x₁ and y₁

$\sqrt{( {x - 0})^{2} + {(y - 0)}^{2} }$

$\sqrt{ {x}^{2} + {y}^{2} } = 2 \sqrt{13}$

${x}^{2} + {y}^{2} = 4 \times 13 = 52$

now we know that :-

• squares are always positive.
• the squares before 52 are : 1,4,9,16,25,36,49
• and sum of any two of these shud be equal to 52

And this time,

we can see that 16+36 = 52 and there is no other sum possible.

so, x must be 4 or 6

and y must be 6 or 4

but we also know that the point should exist on the same line as of origin and it's corresponding point.

so, the slope must be equal⠀⠀⠀⠀

i.e 2/3 = x₁/y₁

and we can see that 4/6 = 2/3

and 6/4 ≠ 2/3⠀⠀

so, x₁ = 4

& y₁ = 6

Similarly we can find other vertices.

$\sqrt{ ( {4})^{2} + ( { - 4})^{2} }$

$4 \sqrt{2}$

so , the distance of (x₂ , y₂) from origin will be double of it i.e 8√2 .

below x and y represents x₂ and y₂

$\sqrt{ {x}^{2} + {y}^{2} } = 8 \sqrt{2}$

${x}^{2} + {y}^{2} = 64 \times 2$

x₂ = 8 , -8

y₂ = -8 , 8

Now, y of the original point is negative so, the vertex of the dilated one will have y in negative too.

therefore : x₂ = 8 & y₂ = -8

Similarly we can find (x₃, y₃)

An Easy Method :

Whatever the scale factor is just multiply it with the vertices of the current triangle. it will give you

the vertices of the triangle that will be formed

(2,3)——— ×2———»(4,6)

(4,-4)———×2———»(8,-8)

(2,-2)———×2———»(4,-4)

We can also approach the answer by :-

1. Finding 2 points and then using the formula of area of triangle. {as we know area will be quadrupled}

2. form the eq. of line(s) of new triangle and use the parallel configuration of coefficients.

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