A jogger starts running from a point 
A
A with coordinates 
(
0
,
0
)
(0,0) and reaches point 
B
B with coordinates 
(
3
,
9
)
(3,9). He turns 90 degrees to the right relative to his initial direction and continues running in a straight line at the same distance as he ran between point A and B, reaching and stopping at point 
C
C. What is the final coordinate of point 
C
C?

Let's break down the problem step by step to find the coordinates of point \( C \).

### Step 1: Find the Distance Between Points A and B
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, point \( A \) has coordinates \((0, 0)\), and point \( B \) has coordinates \((3, 9)\). So the distance \( AB \) is:

\[
AB = \sqrt{(3 - 0)^2 + (9 - 0)^2} = \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
\]

### Step 2: Determine the Direction of the Jogger's Turn
The jogger turns 90 degrees to the right from the direction of \( AB \). The direction from \( A \) to \( B \) forms a vector:

\[
\overrightarrow{AB} = (3 - 0, 9 - 0) = (3, 9)
\]

When the jogger turns 90 degrees to the right, the new direction vector will be perpendicular to \(\overrightarrow{AB}\). To find a perpendicular vector, we can swap the components and change the sign of one of them. The perpendicular vector to \( (3, 9) \) could be \( (9, -3) \) or \( (-9, 3) \).

Since the jogger turns to the right, we take the vector \( (9, -3) \) as the direction vector for \( BC \).

### Step 3: Find the Coordinates of Point C
We need to move from point \( B \) in the direction of \( (9, -3) \) for the same distance \( AB = 3\sqrt{10} \).

The direction vector \( (9, -3) \) has a magnitude of:

\[
\text{Magnitude of } (9, -3) = \sqrt{9^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}
\]

Since the magnitude is exactly \( 3\sqrt{10} \), we can directly add this vector to the coordinates of point \( B \):

\[
C = B + (9, -3) = (3 + 9, 9 - 3) = (12, 6)
\]

### Final Answer
The coordinates of point \( C \) are \( \boxed{(12, 6)} \).

---

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What would be the coordinates of point \( C \) if the jogger turned 90 degrees to the left?
2. How do you calculate the midpoint between points \( A \) and \( C \)?
3. What would the jogger's final coordinates be if he continued running twice the distance between points \( A \) and \( B \)?
4. How can we calculate the angle between the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \)?
5. If point \( A \) were at \( (1, 2) \) instead of \( (0, 0) \), what would be the coordinates of point \( C \)?
6. How would the problem change if the jogger turned 180 degrees instead of 90 degrees?
7. What is the area of the triangle formed by points \( A \), \( B \), and \( C \)?
8. How does the concept of vector addition apply to this problem?

**Tip:** When finding a vector perpendicular to another, you can simply swap the components and change the sign of one component.

Explore how to determine the final coordinates of point C after a jogger starts at point A (0, 0), reaches point B (3, 9), and turns 90 degrees to the right. Learn about vectors, the distance formula, and geometric calculations involved. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation