Analytical geometry, help with this

Welcome back Kwanga — missed you this week. Here are the solutions to your analytical geometry problems. 2.1 Determine the coordinates of E. E is the midpoint of line segment AB. The coordinates of A are $(5, 0)$ and B are $(7, 6)$. The midpoint formula is: $$E = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ Step 1: Substitute the coordinates of A and B into the midpoint formula. $$E = \left(\frac{5+7}{2}, \frac{0+6}{2}\right)$$ Step 2: Calculate the coordinates. $$E = \left(\frac{12}{2}, \frac{6}{2}\right)$$ $$E = (6, 3)$$ The coordinates of E are $\boxed{(6, 3)}$. 2.2 Determine the equation of line BA. To find the equation of a line, we need its slope and a point on the line. The coordinates of A are $(5, 0)$ and B are $(7, 6)$. Step 1: Calculate the slope ($m$) of line BA using the formula $m = \frac{y_2-y_1}{x_2-x_1}$. $$m_{BA} = \frac{6-0}{7-5}$$ $$m_{BA} = \frac{6}{2}$$ $$m_{BA} = 3$$ Step 2: Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, with point A$(5, 0)$ and slope $m_{BA}=3$. $$y - 0 = 3(x - 5)$$ Step 3: Simplify the equation. $$y = 3x - 15$$ The equation of line BA is $\boxed{y = 3x - 15}$. 2.3 Line BA is parallel to straight line with equation: $rx - 3y + 5 = 0$. Calculate the value of r. If two lines are parallel, their slopes are equal. Step 1: The slope of line BA is $m_{BA} = 3$ (from 2.2). Step 2: Find the slope of the given line $rx - 3y + 5 = 0$ by rearranging it into the slope-intercept form $y = mx + c$. $$rx - 3y + 5 = 0$$ $$-3y = -rx - 5$$ $$3y = rx + 5$$ $$y = \frac{r}{3}x + \frac{5}{3}$$ The slope of this line is $m_2 = \frac{r}{3}$. Step 3: Equate the slopes since the lines are parallel. $$m_{BA} = m_2$$ $$3 = \frac{r}{3}$$ Step 4: Solve for $r$. $$r = 3 \times 3$$ $$r = 9$$ The value of r is $\boxed{9}$. Send me the next one 📸