5. What are the coordinates of $$ Y $$ if $$ (5,12) $$ is $$ \frac{1}{3} $$ of the way from $$ X $$ to $$ Y $$ ?

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UpStudy AI · Accepted Answer

To determine the coordinates of point $$ Y $$ given that $$ (5, 12) $$ is $$ \frac{1}{3} $$ of the way from $$ X $$ to $$ Y $$, we can use the concept of **section formula** in coordinate geometry.

### Step-by-Step Solution:

1. **Understanding the Problem:**
   
   - Let $$ X = (x_1, y_1) $$ and $$ Y = (x_2, y_2) $$.
   - The point $$ (5, 12) $$ divides the line segment $$ XY $$ in the ratio $$ 1:2 $$ (since it's $$ \frac{1}{3} $$ of the way from $$ X $$ to $$ Y $$).

2. **Applying the Section Formula:**
   
   The section formula states that a point $$ P $$ dividing the line segment $$ XY $$ in the ratio $$ m:n $$ has coordinates:
   $$
   P = \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
   $$
   
   Here, $$ P = (5, 12) $$, $$ m = 1 $$, and $$ n = 2 $$.

3. **Setting Up Equations:**
   
   Applying the section formula:
   $$
   (5, 12) = \left( \frac{1 \cdot x_2 + 2 \cdot x_1}{1 + 2}, \frac{1 \cdot y_2 + 2 \cdot y_1}{1 + 2} \right)
   $$
   
   Simplifying:
   $$
   5 = \frac{x_2 + 2x_1}{3} \quad \text{and} \quad 12 = \frac{y_2 + 2y_1}{3}
   $$
   
   Multiply both sides by 3:
   $$
   x_2 + 2x_1 = 15 \quad \text{and} \quad y_2 + 2y_1 = 36
   $$

4. **Assuming $$ X $$ is the Origin:**
   
   If $$ X $$ is not given, a common assumption is that $$ X $$ is the origin $$ (0, 0) $$. Plugging $$ X = (0, 0) $$ into the equations:
   $$
   x_2 + 2(0) = 15 \Rightarrow x_2 = 15
   $$
   $$
   y_2 + 2(0) = 36 \Rightarrow y_2 = 36
   $$
   
5. **Conclusion:**
   
   Therefore, the coordinates of $$ Y $$ are $$ (15, 36) $$.

### Final Answer:
The coordinates of $$ Y $$ are $$ \boxed{(15,\ 36)} $$.
The coordinates of $$ Y $$ are $$ (15, 36) $$.

What are the coordinates of if
is of the way from to ?

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