Question

team task gallery walk! briah valentine is designing a park using grid - paper. she wants to build a sidewalk that connects with the fountain at $p(0,1)$ and is perpendicular to the existing sidewalk that passes through points $q(-6,-2)$ and $r(0,-6)$. graph the line that represents the new sidewalk.

Explanation:

Step1: Find slope of existing sidewalk

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $Q(-6,-2)$ and $R(0,-6)$, we have $x_1=-6,y_1 = - 2,x_2=0,y_2=-6$. Then $m=\frac{-6-(-2)}{0 - (-6)}=\frac{-6 + 2}{6}=\frac{-4}{6}=-\frac{2}{3}$.

Step2: Find slope of new sidewalk

If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the new sidewalk be $m_1$. Since $m\times m_1=-1$ and $m =-\frac{2}{3}$, then $-\frac{2}{3}m_1=-1$, so $m_1=\frac{3}{2}$.

Step3: Use point - slope form to find equation of new sidewalk

The point - slope form of a line is $y - y_0=m_1(x - x_0)$, where $(x_0,y_0)$ is a point on the line. The new sidewalk passes through $P(0,1)$, so $x_0 = 0,y_0=1$ and $m_1=\frac{3}{2}$. The equation is $y - 1=\frac{3}{2}(x - 0)$, which simplifies to $y=\frac{3}{2}x+1$.

Step4: Graph the line

To graph $y=\frac{3}{2}x + 1$, first plot the y - intercept which is the point $(0,1)$. Then, using the slope $\frac{3}{2}$, from the point $(0,1)$ move 2 units to the right and 3 units up to get another point $(2,4)$. Draw a straight line passing through these two points.

Answer:

Graph the line $y=\frac{3}{2}x + 1$ by plotting the y - intercept $(0,1)$ and using the slope $\frac{3}{2}$ to find another point $(2,4)$ and then drawing a line through them.