Determine whether the polygons with the given vertices are similar. Support your answer by describing a transformation.
V(3,2) W(8,2) X(1,5)
R(6,4) S(16,4) T(3,15)
I think they are not similar because V and W are dilated by 2 and X is dilated by 3. They need to be dilated by the same factor to be similar.
because if they were similar
then if V to W is 5
and R to S is 10
then if similar
W to X must be HALF of S to T
It is not, it is sqrt 5 times
I know which sides to compare because I drew a quick sketch of the two figures.
agree
V to W = sqrt (25+0) = 5
R to S = sqrt (100+0) = 10
W to X =sqrt(49+9) = sqrt(58)
S to T = sqrt(169+121) = sqrt (5*58) oh my :(, not twice, would be sqrt 4*58 if similar
V, W, and X are part of one polygon, and R, S, T are part of another polygon. So why is V to W calculated?
To determine whether the polygons with the given vertices are similar, we need to determine if there exists a transformation that can map one polygon onto the other. One possible transformation to consider is dilation.
Dilation is a transformation that changes the size of a shape by either expanding or contracting it. It preserves the shape's proportions but changes its size. If two polygons can be mapped onto each other by dilation, they are considered similar.
To check if the polygons are similar, we need to compare the corresponding side lengths of both polygons. If the ratios of corresponding side lengths are equal, then the polygons are similar.
Looking at the vertices:
Polygon 1: V(3,2), W(8,2), X(1,5)
Polygon 2: R(6,4), S(16,4), T(3,15)
Let's calculate the lengths of the sides of each polygon:
Polygon 1:
- Side VW: √[(8 - 3)^2 + (2 - 2)^2] = √(5^2) = 5
- Side VX: √[(1 - 3)^2 + (5 - 2)^2] = √((-2)^2 + 3^2) = √(4 + 9) = √13
- Side WX: √[(1 - 8)^2 + (5 - 2)^2] = √((-7)^2 + 3^2) = √(49 + 9) = √58
Polygon 2:
- Side RS: √[(16 - 6)^2 + (4 - 4)^2] = √(10^2) = 10
- Side RT: √[(3 - 6)^2 + (15 - 4)^2] = √((-3)^2 + 11^2) = √(9 + 121) = √130
- Side ST: √[(3 - 16)^2 + (15 - 4)^2] = √((-13)^2 + 11^2) = √(169 + 121) = √290
Now, let's compare the ratios of the corresponding sides:
Ratio of corresponding sides VW and RS: 5/10 = 1/2
Ratio of corresponding sides VX and RT: √13/√130 = 1/√10
Ratio of corresponding sides WX and ST: √58/√290 = 1/√5
As we can see, the ratios of the corresponding sides are not equal. Therefore, the polygons with the given vertices are not similar.
It is worth noting that in this case, dilation is not the only possible transformation to consider. Other transformations such as translation, rotation, or reflection could also be explored. However, for this specific question, we focused on dilation as the requested transformation.