Let point P be the weighted average of points  A(7,3)
and  B(–3,–10)
, where point A weighs three times as much as point B. What are the coordinates of point P?

Multiple choice question.

A)
(−2,−27/4)

B)
(2,−7/2)

C)
(9/2,−1/4)

D)
(6,−1/3)

To find the coordinates of point \( P \), which is the weighted average of points \( A(7, 3) \) and \( B(-3, -10) \), with point \( A \) weighing three times as much as point \( B \), we use the formula for the weighted average of two points:

\[
P = \left( \frac{m_A \cdot x_A + m_B \cdot x_B}{m_A + m_B}, \frac{m_A \cdot y_A + m_B \cdot y_B}{m_A + m_B} \right)
\]

Here, \( m_A = 3 \) and \( m_B = 1 \), since point \( A \) weighs three times as much as point \( B \).

Let's calculate the coordinates of \( P \):

1. **x-coordinate of P**:

\[
x_P = \frac{3 \cdot 7 + 1 \cdot (-3)}{3 + 1} = \frac{21 - 3}{4} = \frac{18}{4} = \frac{9}{2}
\]

2. **y-coordinate of P**:

\[
y_P = \frac{3 \cdot 3 + 1 \cdot (-10)}{3 + 1} = \frac{9 - 10}{4} = \frac{-1}{4}
\]

So, the coordinates of point \( P \) are \( \left( \frac{9}{2}, -\frac{1}{4} \right) \).

The correct answer is:

**C) \( \left(\frac{9}{2}, -\frac{1}{4}\right) \)**

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Would you like further details or have any questions? Here are five related questions:

1. How do you find the midpoint of two points?
2. What is the general formula for the weighted average of three points?
3. How can we find the centroid of a triangle given its vertices?
4. What is the significance of the weights in calculating the weighted average?
5. How do you apply the distance formula between two points?

**Tip**: Always double-check your calculations when dealing with fractions to avoid simple arithmetic errors.

Learn how to calculate the weighted average of points A(7,3) and B(-3,-10) where point A weighs three times as much as point B. Understand the concept of weighted averages and apply it to solve coordinate geometry problems.

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