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**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**.

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**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**: \( A'(-4, -2), B'(-1, -1), C'(-2, 2) \).

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**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.

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**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**: \( 2 + 7 = 9 \).

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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.

What is the y-coordinate of the midpoint of AB? A triangle with coordinates A(1, 1), B(4, 2), C(3, 5) is translated three units down and five units to the left. What are the coordinates of the new triangle? Demari drew oblique segment AB on a coordinate grid and translated it 4 units to the left to form A'B'. Which statement must be true? E'Naiya conjectures that the sum of any two prime numbers is even. Which of the following is a counterexample that shows E'Naiya's conjecture is false?

Solve problems involving midpoints, triangle translations, and prime numbers. Learn how to translate triangles in coordinate geometry, find midpoints between two points, and identify a counterexample that disproves a conjecture about prime numbers. Suitable for students in Grades 7-9.

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