Question

line gh is represented by the equation $y = \frac{5}{4}x + \frac{67}{4}$. line pv is perpendicular to line gh. the coordinates of point p are (5, 15). determine the equation of line pv in slope - intercept form. \
slope of line gh $m_1$ \
slope of line pv $m_2$ \
point - slope form of line pv $y - y_1 = m(x - x_1)$ \
(there is also a coordinate grid with points g, h, p plotted)

Explanation:

Step1: Find slope of GH ($m_1$)

The equation of line GH is $y = \frac{5}{4}x + \frac{67}{4}$. In slope - intercept form $y=mx + b$, the slope $m_1=\frac{5}{4}$.

Step2: Find slope of PV ($m_2$)

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of PV be $m_2$. Then $m_1\times m_2=-1$.
Substitute $m_1 = \frac{5}{4}$ into the equation: $\frac{5}{4}\times m_2=-1$.
Solve for $m_2$: $m_2=-\frac{4}{5}$.

Step3: Use point - slope form for PV

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(5,15)$ and $m = m_2=-\frac{4}{5}$.
Substitute the values: $y - 15=-\frac{4}{5}(x - 5)$.

Step4: Convert to slope - intercept form

Expand the right - hand side: $y-15 = -\frac{4}{5}x+4$.
Add 15 to both sides: $y=-\frac{4}{5}x + 4 + 15$.
Simplify: $y=-\frac{4}{5}x+19$.

Answer:

• Slope of Line GH ($m_1$): $\frac{5}{4}$
• Slope of Line PV ($m_2$): $-\frac{4}{5}$
• Point - Slope Form of Line PV: $y - 15=-\frac{4}{5}(x - 5)$
• Slope - Intercept Form of Line PV: $y=-\frac{4}{5}x + 19$