Question

name:
geometry unit 1 rigid transformation quiz (10 pts)
date:
period:
1.
a. translate triangle abc by the directed line - segment from c to b. label the new images with a single - prime notation ( ).
b. rotate triangle abc clockwise 180 degrees around point c. label the new images with a double - prime notation ( ).
c. reflect triangle abc along the line m. label the new images with a triple - prime notation ( ).
d. triangle abc is a right triangle. make a conjecture about the figure created by these 3 transformations. be specific.

Explanation:

Step1: Translation

To translate triangle $ABC$ by the directed line - segment from $C$ to $H$, we move each vertex of $\triangle ABC$ the same distance and in the same direction as the vector $\overrightarrow{CH}$. Let the coordinates of $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ and $H(x_4,y_4)$. The new coordinates of $A'$, $B'$, $C'$ are obtained by adding the components of the vector $\overrightarrow{CH}=(x_4 - x_3,y_4 - y_3)$ to the coordinates of $A$, $B$, $C$ respectively.

Step2: Rotation

To rotate $\triangle ABC$ clock - wise $180$ degrees around point $C$, we use the rotation formula for a point $(x,y)$ around a center $(a,b)$ by an angle $\theta = 180^{\circ}$. The formula is $(x',y')=(a-(x - a),b-(y - b))$. For each vertex of $\triangle ABC$ with respect to point $C$, we apply this formula to get the double - prime vertices.

Step3: Reflection

To reflect $\triangle ABC$ along the line $m$, we use the reflection formula. For a point $(x,y)$ and a line $y=mx + c$, the reflected point $(x',y')$ can be found by first finding the perpendicular distance from the point to the line, and then moving the same distance on the other side of the line. In the case of a vertical or horizontal line (if line $m$ is one of them), the reflection is simpler. For example, if $m$ is a vertical line $x = k$, the $x$ - coordinate of a point $(x,y)$ is transformed as $x'=2k - x$ and the $y$ - coordinate remains the same.

Step4: Conjecture

Since translations, rotations, and reflections are rigid motions, the figure created by these three transformations will be congruent to the original right - triangle $\triangle ABC$. The combined figure (the original and the three transformed triangles) may form a symmetric figure. For example, if the line of reflection and the direction of translation and rotation center are in certain positions, the overall figure may have rotational or reflectional symmetry.

Answer:

a. Follow the translation rule to get $\triangle A'B'C'$.
b. Use the rotation formula around point $C$ to get $\triangle A''B''C''$.
c. Apply the reflection formula across line $m$ to get $\triangle A'''B'''C'''$.
d. The figure formed by the original and transformed triangles is congruent to $\triangle ABC$ and may have symmetry properties.