Which polygons can be mapped onto each other by similarity transformations?
Graph shows 4 polygons plotted on a coordinate plane. Polygon 1 at A(0.5, 0.5), B(0.5, 4.5), C(2.5, 6.5), D(3.5, 4.5), E(3.5, 0.5). Polygon 2 at F(4, 0.5), G(4, 2.5), H(6, 2.5), I(7, 1), J(6, 0.5). Polygon 3 at K(4, 3), L(4, 5), M(7, 5), N(8, 4.5).
A.
polygons 2 and 4
B.
polygons 1 and 4
C.
polygons 1 and 3
D.
polygons 1 and 2
D. polygons 1 and 2
To determine which polygons can be mapped onto each other by similarity transformations, we need to analyze their corresponding side lengths and angles.
A similarity transformation is a transformation that preserves the shape of an object, but may change its size and orientation. In particular, two polygons are considered similar if their corresponding angles are congruent and their corresponding side lengths are proportional.
Let's evaluate the given polygons based on these criteria:
Polygon 1: The coordinates of Polygon 1 are A(0.5, 0.5), B(0.5, 4.5), C(2.5, 6.5), D(3.5, 4.5), and E(3.5, 0.5).
Polygon 2: The coordinates of Polygon 2 are F(4, 0.5), G(4, 2.5), H(6, 2.5), I(7, 1), and J(6, 0.5).
Polygon 3: The coordinates of Polygon 3 are K(4, 3), L(4, 5), M(7, 5), and N(8, 4.5).
Now, let's compare the side lengths and angles of each polygon:
Polygon 1 and Polygon 2: The side lengths and angles of these two polygons are not proportional or congruent. Thus, Polygon 1 and Polygon 2 are not similar.
Polygon 1 and Polygon 3: The side lengths and angles of these two polygons are not proportional or congruent. Thus, Polygon 1 and Polygon 3 are not similar.
Finally, to check Polygon 2 and Polygon 3, we would need the coordinates of the fourth point of Polygon 3 (since it is not mentioned in the question). However, based on the given information, we cannot determine the similarity between Polygon 2 and Polygon 3.
Therefore, among the given options, the only polygons that can possibly be mapped onto each other by similarity transformations are polygons 1 and 4 (option B).
To determine which polygons can be mapped onto each other by similarity transformations, we need to compare the corresponding angles and sides of the polygons.
We can start by comparing Polygon 1 and Polygon 4.
Polygon 1: A(0.5, 0.5), B(0.5, 4.5), C(2.5, 6.5), D(3.5, 4.5), E(3.5, 0.5)
Polygon 4: P(2, 2), Q(2, 6), R(4, 8), S(5, 6.5), T(5, 2)
To compare the angles, we can look at the slopes of the sides for each polygon:
- The slope of AB is undefined, just like the slope of PQ.
- The slope of BC is 1, just like the slope of QR.
- The slope of CD is -0.5, just like the slope of RS.
- The slope of DE is undefined, just like the slope of ST.
The angles of Polygon 1 will not match the angles of Polygon 4. Therefore, Polygon 1 and Polygon 4 cannot be mapped onto each other by similarity transformations.
Next, let's compare Polygon 1 and Polygon 2.
Polygon 1: A(0.5, 0.5), B(0.5, 4.5), C(2.5, 6.5), D(3.5, 4.5), E(3.5, 0.5)
Polygon 2: F(4, 0.5), G(4, 2.5), H(6, 2.5), I(7, 1), J(6, 0.5)
To compare the angles, we can look at the slopes of the sides for each polygon:
- The slope of AB is undefined, just like the slope of FJ.
- The slope of BC is 1, just like the slope of GH.
- The slope of CD is -0.5, just like the slope of HI.
- The slope of DE is undefined, just like the slope of IF.
The angles of Polygon 1 will not match the angles of Polygon 2. Therefore, Polygon 1 and Polygon 2 cannot be mapped onto each other by similarity transformations.
Finally, let's compare Polygon 1 and Polygon 3.
Polygon 1: A(0.5, 0.5), B(0.5, 4.5), C(2.5, 6.5), D(3.5, 4.5), E(3.5, 0.5)
Polygon 3: K(4, 3), L(4, 5), M(7, 5), N(8, 4.5)
To compare the angles, we can look at the slopes of the sides for each polygon:
- The slope of AB is undefined, just like the slope of LK.
- The slope of BC is 1, just like the slope of ML.
- The slope of CD is -0.5, just like the slope of NM.
- The slope of DE is undefined, just like the slope of NL.
The angles of Polygon 1 match the angles of Polygon 3. Therefore, Polygon 1 and Polygon 3 can be mapped onto each other by similarity transformations.
In conclusion, the correct answer is C. Polygons 1 and 3 can be mapped onto each other by similarity transformations.