Euclidean geometry: transformations and congruence

High School Math 1 · Unit 16 · transformations-congruence · Teacher edition

transformations-congruence

Name ______________________________Date ______________

1.
The point $(3, -2)$ is rotated $90^\circ$ counter-clockwise about the origin. Enter the $x$ -coordinate of the image.
Answer: ______________
2.
A vertex $P(-4, 6)$ of a triangle is rotated $90^\circ$ counter-clockwise about the origin, and the result is then dilated by a scale factor of $\dfrac{1}{2}$ about the origin. Enter the $y$ -coordinate of the final image.
Answer: ______________
3.
A single transformation maps $A(2, 5)$ to $A'(5, 2)$ and $B(-1, 3)$ to $B'(3, -1)$ . Which transformation is it?
1. (A)
  Reflection over the $x$ -axis
2. (B)
  Rotation of $90^\circ$ counter-clockwise about the origin
3. (C)
  Translation by $(3, -3)$
4. (D)
  Reflection over the line $y = x$
4.
Two triangles share some equal measurements. Which set of equal parts guarantees the triangles are congruent?
1. (A)
  Two sides and a non-included angle
2. (B)
  Three angles
3. (C)
  Two sides and the included angle
4. (D)
  One side and one angle
5.
A triangle has vertices $(1, 1)$ , $(5, 1)$ , and $(1, 4)$ . It is dilated by a scale factor of $3$ about the origin. What is the perimeter of the image, in units?
Answer: ______________

Answer key — Euclidean geometry: transformations and congruence

1.
2transformations-congruence-01
$(3, -2) \mapsto (-(-2), 3) = (2, 3)$ . The $x$ -coordinate is $\mathbf{2}$ . (The rule $(x, y) \mapsto (-x, y)$ is a reflection over the $y$ -axis, not a rotation.)
2.
-2transformations-congruence-02
Rotation: $(-4, 6) \mapsto (-6, -4)$ . Dilation: $(-6, -4) \mapsto (-3, -2)$ . The $y$ -coordinate is $\mathbf{-2}$ . The rotation keeps the triangle congruent; the dilation makes the final image similar to the original, not congruent.
3.
(D)
Reflection over the line $y = x$
transformations-congruence-03
$(x, y) \mapsto (y, x)$ is the reflection over $y = x$ , and it fits both points. (Reflection over the $x$ -axis gives $(2, -5)$ ; the $90^\circ$ rotation gives $(-5, 2)$ ; the translation by $(3, -3)$ fits $A$ but sends $B$ to $(2, 0)$ .)
4.
(C)
Two sides and the included angle
transformations-congruence-04
Two sides and the included angle is SAS, a valid congruence shortcut. (SSA is not — two different triangles can share those parts; AAA only gives similarity; one side and one angle is far too little.)
5.
36transformations-congruence-05
Original perimeter $= 3 + 4 + 5 = 12$ . Scale factor $3$ : image perimeter $= 3 \cdot 12 = \mathbf{36}$ . (Multiplying by $9$ would be the area rule; the image's area is $9 \cdot 6 = 54$ .)