Question

directions: use this information to answer parts a, b, and c.
the endpoints of one side of square ( pqrs ) are at ( p(1,1) ) and ( q(3,4) ).
part a
what is the equation for the line that contains ( overline{qr} )?
enter the correct expression in the box.
show hints

Explanation:

Step1: Find the slope of \( \overline{PQ} \)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For points \( P(1,1) \) and \( Q(3,4) \), \( m_{PQ}=\frac{4 - 1}{3 - 1}=\frac{3}{2} \).
Since \( \overline{PQ}\perp\overline{QR} \) (adjacent sides of a square are perpendicular), and the product of slopes of perpendicular lines \( m_1\times m_2=- 1 \). Let \( m_{QR}=m \), then \( \frac{3}{2}\times m=-1 \), so \( m =-\frac{2}{3} \).

Step2: Use the point - slope form \( y - y_1=m(x - x_1) \)

We know that the line passes through \( Q(3,4) \) and \( m =-\frac{2}{3} \). Substitute \( x_1 = 3,y_1 = 4,m=-\frac{2}{3} \) into \( y - y_1=m(x - x_1) \).
\( y - 4=-\frac{2}{3}(x - 3) \)
Expand: \( y-4=-\frac{2}{3}x + 2 \)
Add 4 to both sides: \( y=-\frac{2}{3}x+6 \)

Answer:

\( y =-\frac{2}{3}x + 6 \)