Question

75 points to whoever does all of this for me.

1. Draw, in black, the triangle with vertices (-3, 2), (1, 1), and (2, 5).

2. Draw the triangle's image under the dilation (x, y ) → (2x, 2y).

3. Explain clearly why the two triangles must be similar.

4. How could a dilation create a figure which is congruent to the original one? [Hint: there are two ways.]


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Answer 1

Whenever an image goes under dilation we multiply the co-ordinates of each vertex with a scale factor (k).

Points (x,y) becomes (2x,2y) so the scale factor is 2.

Explanation:

1.The diagram is shown below.

2.The co-ordinates (-3,2),(1,1) and (2,5) will become (-6,4),(2,2) and (4,10).

3.Dilated figures are similar.

The two triangles must be similar as we can see
`(AC)/(ZY)` =
`(BC)/(XY)` = 2

We can find the length of AC,ZY,BC and XY by using distance formula.

Distance formula =
`√((x2-x1)^2+(y2-y1)^2)`

AC=
`√((4-2)^2+(10-2)^2)` = Approx 8

BC=
`√((-6-2)^2+(4-2)^2)` =Approx 8

For triangle XYZ

ZY=
`√((2-1)^2+(5-1)^2)`=Approx 4

XY=
`√((-3-1)^2+(2-1)^2)`=Approx 4

`(AC)/(ZY)` =
`(BC)/(XY)` = 2

Similar triangles must have proportional side lengths.

Statement:Two figures are said to be similar when one figure can be obtained from the other by a single transformation that includes translation, reflection, rotation and dilation,the size may not be same.

So in above question the triangles are undergoing dilation with a positive scale factor,hence both are similar.

4.Dilation and congruence.

When the scale factor is equivalent to one (1).

The dilation create a figure which is congruent to the original one.

Similarity preserves shape, but not necessarily size making the figures similar.It is possible for similar figures to have a scale factor of 1 then it can be said that all congruent figures are also similar.