Problem 4: Equation of a Line Perpendicular to a Given Line

We are tasked with finding the equation of a line passing through the point $(22, 7)$ and perpendicular to a line passing through the points $(26, 1)$ and $(0, 4)$.

1. Find the slope of the given line passing through $(26, 1)$ and $(0, 4)$. The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the values: $m = \frac{4 - 1}{0 - 26} = \frac{3}{-26} = -\frac{3}{26}$ The slope of the line through $(26, 1)$ and $(0, 4)$ is $-\frac{3}{26}$.

2. Find the slope of the perpendicular line. The slope of a perpendicular line is the negative reciprocal of the original slope. If the slope of the original line is $-\frac{3}{26}$, the slope of the perpendicular line is: $m_{\perp} = \frac{26}{3}$

3. Write the equation of the perpendicular line using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is $(22, 7)$ and $m_{\perp} = \frac{26}{3}$: $y - 7 = \frac{26}{3}(x - 22)$ Simplify the equation: $y - 7 = \frac{26}{3}x - \frac{26}{3} \times 22$ $y - 7 = \frac{26}{3}x - \frac{572}{3}$ Add 7 to both sides: $y = \frac{26}{3}x - \frac{572}{3} + 7$ Convert 7 to have a denominator of 3: $y = \frac{26}{3}x - \frac{572}{3} + \frac{21}{3}$ $y = \frac{26}{3}x - \frac{551}{3}$ Therefore, the equation of the line perpendicular to the given line and passing through the point $(22, 7)$ is: $y = \frac{26}{3}x - \frac{551}{3}$

Problem 5: Equation of a Line Perpendicular to a Given Line

We are tasked with finding the equation of a line that passes through the point $(24, 4)$ and is perpendicular to a line that intersects at the point $(5, 9)$.

1. Find the slope of the given line. We do not have a second point, but the problem mentions perpendicular lines. We assume the line's slope is unknown, but we'll denote it as $m$. If the equation of the first line were available, we'd find the slope of the perpendicular line using the negative reciprocal.

2. Using the point-slope form, we have: $y - y_1 = m_{\perp}(x - x_1)$ We need more information about the given line’s slope (or points on the line).

Would you like more details or to clarify the second question?

Here are five related questions:

1. How do you find the slope between two points on a line?
2. What is the negative reciprocal of a number?
3. How do you convert point-slope form to slope-intercept form?
4. What is the relationship between perpendicular lines?
5. How do you handle horizontal and vertical lines when finding perpendicular slopes?

Tip: The slope of a line is undefined when it is vertical, and its perpendicular line will be horizontal.