Question

triangle qrs is graphed on a coordinate plane with vertices at (-3, 3), (2, 7), and (6, 2), as shown. the triangle is then reflected over the y - axis to form △qrs and then translated down 3 units to form △qrs. which of the following are true? the area of qrs is the same as the area of qrs. point s is located in quadrant iii. point q is located in quadrant ii. line segment rs is twice as long as line segment rs. angles q, q, q are congruent.

Explanation:

Step1: Understand transformation properties

Reflections and translations are rigid - motions. Rigid - motions preserve shape and size, including area and angle measures.

Step2: Analyze area

Since reflections and translations are rigid - motions, the area of $\triangle Q''R''S''$ is the same as the area of $\triangle QRS$.

Step3: Find coordinates after transformations

The rule for reflecting a point $(x,y)$ over the $y$ - axis is $(-x,y)$. The vertices of $\triangle QRS$ are $Q(-3,3)$, $R(2,7)$, $S(6,2)$. After reflecting over the $y$ - axis, $Q'(3,3)$, $R'(-2,7)$, $S'(-6,2)$. After translating down 3 units, $Q''(3,0)$, $R''(-2,4)$, $S''(-6,-1)$.

Step4: Determine quadrants

Point $S''(-6,-1)$ is in Quadrant III. Point $Q'$ has coordinates $(3,3)$ which is in Quadrant I.

Step5: Analyze line - segment lengths

Rigid - motions preserve lengths, so $R'S'=RS$.

Step6: Analyze angle measures

Rigid - motions preserve angle measures, so $\angle Q=\angle Q'=\angle Q''$.

Answer:

The area of $Q''R''S''$ is the same as the area of $QRS$; Point $S''$ is located in Quadrant III; Angles $Q$, $Q'$, $Q''$ are congruent.