Question

1. on the figure shown, point s is not shown but has coordinates (9,a). if $overrightarrow{rs}perpoverrightarrow{tv}$, then what is the value of a? t (2,4) r (-1,1) v (4,-1)

Explanation:

Step1: Find the slope of $\overleftrightarrow{TV}$

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $T(2,4)$ and $V(4, - 1)$, we have $m_{TV}=\frac{-1 - 4}{4 - 2}=\frac{-5}{2}$.

Step2: Find the slope of $\overleftrightarrow{RS}$

Since $\overleftrightarrow{RS}\perp\overleftrightarrow{TV}$, the product of their slopes is - 1. Let the slope of $\overleftrightarrow{RS}$ be $m_{RS}$. Then $m_{RS}\times m_{TV}=-1$. So $m_{RS}=\frac{2}{5}$.

Step3: Calculate the slope of $\overleftrightarrow{RS}$ using coordinates of $R$ and $S$

For points $R(-1,1)$ and $S(9,a)$, $m_{RS}=\frac{a - 1}{9-(-1)}=\frac{a - 1}{10}$.

Step4: Solve for $a$

Set $\frac{a - 1}{10}=\frac{2}{5}$. Cross - multiply: $5(a - 1)=2\times10$. Expand: $5a-5 = 20$. Add 5 to both sides: $5a=25$. Divide by 5: $a = 5$.

Answer:

$5$