https://file.aiquickdraw.com/mathbot/file/17263630547996daibsxn.jpg

We are asked to find the equation of a line that crosses the point \( (0, 7) \) and is perpendicular to the line passing through the points \( (-2, 4) \) and \( (4, -5) \).

### Step 1: Find the slope of the line passing through points \( (-2, 4) \) and \( (4, -5) \)
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the points \( (-2, 4) \) and \( (4, -5) \):

\[
m = \frac{-5 - 4}{4 - (-2)} = \frac{-9}{6} = -\frac{3}{2}
\]

So, the slope of the line passing through these points is \( -\frac{3}{2} \).

### Step 2: Find the slope of the perpendicular line
The slope of the line perpendicular to this is the negative reciprocal of \( -\frac{3}{2} \). The negative reciprocal is:

\[
m_{\text{perpendicular}} = \frac{2}{3}
\]

### Step 3: Equation of the line using the point-slope form
The line we are looking for passes through the point \( (0, 7) \) and has slope \( \frac{2}{3} \). The equation of a line in point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Substitute \( m = \frac{2}{3} \), \( x_1 = 0 \), and \( y_1 = 7 \):

\[
y - 7 = \frac{2}{3}(x - 0)
\]

Simplifying:

\[
y - 7 = \frac{2}{3}x
\]

\[
y = \frac{2}{3}x + 7
\]

### Step 4: Choose the correct equation
The correct equation is \( y = \frac{2}{3}x + 7 \), which matches **Option D**.

---

Would you like more details on any part of this process?

Here are five related questions to explore:

1. How do you find the equation of a line if given only the slope and one point?
2. What does it mean for two lines to be perpendicular?
3. How would the equation change if the line passed through a different point?
4. Can a vertical line be perpendicular to a non-vertical line? How do we handle such cases?
5. How do you find the midpoint between two points, and does it relate to perpendicular bisectors?

**Tip:** The negative reciprocal rule for slopes of perpendicular lines is key when solving geometry problems involving perpendicularity!

Select the equation for the line crossing the point (0, 7) and perpendicular to the line that passes through the points (-2, 4) and (4, -5).

Learn how to find the equation of a line crossing the point (0,7) that is perpendicular to a line passing through the points (-2,4) and (4,-5). This problem explores key concepts in geometry, including slope calculations and perpendicular line relationships. Suitable for Grades 9-11 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