Let's work through the problems shown in your image:

6) What is the y-coordinate of the midpoint of $AB$?

The midpoint formula between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ For points $A(2, 4)$ and $B(10, 10)$: $\text{Midpoint } = \left( \frac{2+10}{2}, \frac{4+10}{2} \right) = (6, 7)$ Thus, the y-coordinate of the midpoint is 7.

7) A triangle with coordinates $A(1, 1), B(4, 2), C(3, 5)$ is translated three units down and five units to the left.

Translation involves subtracting 3 from the y-coordinates and subtracting 5 from the x-coordinates.

• New $A'$ coordinates: $(1 - 5, 1 - 3) = (-4, -2)$
• New $B'$ coordinates: $(4 - 5, 2 - 3) = (-1, -1)$
• New $C'$ coordinates: $(3 - 5, 5 - 3) = (-2, 2)$

The correct set of coordinates is B: $A'(-4, -2), B'(-1, -1), C'(-2, 2)$.

8) Demari drew oblique segment $AB$ and translated it to form $A'B'$.

A translation moves all points by the same distance and direction, without changing their orientation. Therefore, the two segments must be parallel. The correct answer is B: $AB$ and $A'B'$ are parallel to each other.

9) E'Naiya's conjecture: The sum of any two prime numbers is even.

A counterexample is a pair of prime numbers whose sum is odd. Consider the sum $2 + 7 = 9$, which is odd. This shows the conjecture is false.

The correct answer is B: $2 + 7 = 9$.

Would you like further clarification on any of these problems?

Here are 5 related questions to explore:

1. How do you find the distance between two points in a coordinate plane?
2. What are the properties of triangle translations?
3. How can we verify if two lines are parallel or perpendicular?
4. What are prime numbers, and why is 2 special among them?
5. What is the general rule for finding midpoints between any two points?

Tip: When translating shapes, always apply the translation to each coordinate individually to avoid mistakes.