Question

If $L$ is a line through the points $(2,5)$ and $(4,6)$, what is the value of $k$, so that the point of coordinates $(7,k)$ is on the line $L$?

• A. $0$
• B. $5$
• C. $6$
• D. $\frac{15}{2}$
• E. $11$

Answer 1

Answer: (D)

$\frac{15}{2}$

The slope of the line through $(2,5)$ and $(4,6)$ is given by,

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-5}{4-2}=\frac{1}{2}$

The slope of the line through $(4,6)$ and $(7,k)$ is given by,

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{k-6}{7-4}=\frac{k-6}{3}$

The three points are on the same line and therefore the slopes found above must be equal :

$\frac{1}{2}=\frac{k-6}{3}\Rightarrow 3=2(k-6)\Rightarrow 3=2k-12\Rightarrow 2k=3+12\Rightarrow 2k=15\Rightarrow k=\frac{15}{2}$

Hence, $k=\frac{15}{2}$