Question

The area of the triangle with coordinates $(1,2),(5,5)$ and $(k,2)$ is $15$ square units. Calculate a possible value for $k$.

• A. $-10$
• B. $-9$
• C. $-5$
• D. $5$
• E. $6$

Answer 1

Answer: (B)

$-9$

Area of triangle having vertices $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ is given by
$=\frac{1}{2}\times \left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

Therefore,

Area of triangle is $15=\frac{1}{2}\times |1(5-2)+5(2-2)+k(2-5)|$
$\Rightarrow \frac{1}{2}\times |3+0-3k|=\frac{1}{2}\times |3-3k|=15$
$\Rightarrow |3-3k|=30$ => $|1-k|=10$
$\Rightarrow 1-k=10$ and $1-k=-10$

$\Rightarrow k=-9$ and $k=11$