https://file.aiquickdraw.com/mathbot/file/1727384357323efiuknai.jpg

The question in the image asks:  
**Find the distance between (3, 4) and the line \( y = 2x + 3 \).**

This is a classic problem involving the distance from a point to a line. To solve this, we use the formula for the distance \( d \) between a point \( (x_1, y_1) \) and a line of the form \( Ax + By + C = 0 \):

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

### Step-by-step solution:

1. **Convert the line equation** to the standard form \( Ax + By + C = 0 \).  
   The equation of the line is given as \( y = 2x + 3 \).  
   Rearranging this into the standard form:  
   \[
   2x - y + 3 = 0
   \]
   So, \( A = 2 \), \( B = -1 \), and \( C = 3 \).

2. **Substitute the point** \( (3, 4) \) into the formula. Here, \( x_1 = 3 \) and \( y_1 = 4 \).

\[
d = \frac{|2(3) - 1(4) + 3|}{\sqrt{2^2 + (-1)^2}}
\]

3. **Simplify the expression**:
\[
d = \frac{|6 - 4 + 3|}{\sqrt{4 + 1}} = \frac{|5|}{\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}
\]

Thus, the distance between the point \( (3, 4) \) and the line \( y = 2x + 3 \) is \( \sqrt{5} \).

Would you like more details or have any questions on this? Here are some related questions:

1. What is the formula for finding the distance between two points in space?
2. How can you derive the distance formula from a point to a line geometrically?
3. Can this method be extended to find the distance between a point and a plane in 3D space?
4. What happens to the distance formula if the line is vertical or horizontal?
5. How does the distance between a point and a line change if you move the point?

**Tip:** Always convert the line equation into standard form \( Ax + By + C = 0 \) before applying the distance formula.

Find the distance between (3, 4) and the line y = 2x + 3.

Learn how to find the distance between the point (3, 4) and the line y = 2x + 3 using the distance formula for a point and a line. This problem includes key geometry concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation