Question

question given q(1, - 5), r(4, - 6), s(-6,5), and t(-7,y). find y such that qr⊥st.

Explanation:

Step1: Find slope of QR

Use slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $Q(1,-5)$ and $R(4,-6)$, $m_{QR}=\frac{-6-(-5)}{4 - 1}=\frac{-6 + 5}{3}=-\frac{1}{3}$.

Step2: Find slope of ST

For points $S(-6,5)$ and $T(-7,y)$, $m_{ST}=\frac{y - 5}{-7-(-6)}=\frac{y - 5}{-7 + 6}=5 - y$.

Step3: Use perpendicular - slope relationship

If two lines are perpendicular, the product of their slopes is $- 1$. So $m_{QR}\times m_{ST}=-1$. Substitute the slopes: $-\frac{1}{3}\times(5 - y)=-1$.

Step4: Solve for y

Multiply both sides by $-3$: $5 - y = 3$. Then subtract 5 from both sides: $-y=3 - 5=-2$. Multiply both sides by $-1$ to get $y = 4$.

Answer:

$y = - 4$