Let me go through the questions in the image and solve them step by step.

Question 7:

Translation of points $A(4, 4)$ and $B(2, -4)$ by the vector $T = \begin{bmatrix} 3 \\ 5 \end{bmatrix}$.

The formula for translation is: $A'(x', y') = (x + 3, y + 5)$ Let's apply this:

• For $A(4, 4)$: $A'(x', y') = (4 + 3, 4 + 5) = (7, 9)$
• For $B(2, -4)$: $B'(x', y') = (2 + 3, -4 + 5) = (5, 1)$

So the new coordinates are: $A'(7, 9) \quad \text{and} \quad B'(5, 1)$

Thus, the correct option for Question 7 is a..

Question 8:

Reflection of point $P(5, 6)$ with respect to the line $y = x$.

For a reflection over the line $y = x$, the formula is: $P'(x', y') = (y, x)$ Simply swap the coordinates:

For $P(5, 6)$, the reflection is: $P'(x', y') = (6, 5)$

Thus, the correct option for Question 8 is d..

Question 9:

Rotation of triangle $ABC$ with vertices $A(2, 2)$, $B(4, 5)$, and $C(6, -1)$ by $90^\circ$ counterclockwise around the origin $O(0, 0)$.

The formula for a $90^\circ$ counterclockwise rotation is: $A'(x', y') = (-y, x)$ Apply this for each vertex:

• For $A(2, 2)$: $A'(x', y') = (-2, 2)$
• For $B(4, 5)$: $B'(x', y') = (-5, 4)$
• For $C(6, -1)$: $C'(x', y') = (1, 6)$

Thus, the new coordinates are: $A'(-2, 2), \quad B'(-5, 4), \quad C'(1, 6)$

The correct option for Question 9 is c..

Summary of Correct Answers:

• Question 7: a.
• Question 8: d.
• Question 9: c.

Let me know if you'd like further details or clarifications!

Related questions:

1. How do you reflect points over lines other than $y = x$?
2. What is the general formula for a translation by a vector $T(x, y)$?
3. How do you rotate points around a different center of rotation, not $(0, 0)$?
4. What is the effect of a $180^\circ$ rotation on points?
5. Can you explain the reflection of a triangle over the $x$-axis?

Tip:

When rotating a point by $90^\circ$ counterclockwise, swap the coordinates and negate the new x-coordinate!