Question

If the area of the triangle with vertices $(2,5),(7,k)$ and $(3,1)$ is $10$, then find the value of $k$.

• A. $-5$ or $35$
• B. $5$ or $-35$
• C. $15$ or $-5$
• D. $-5$ or $-25$

Answer 1

Answer: (B)

$5$ or $-35$

If $(x_1,y_1),(x_2,y_2)$ ans $(x_3,y_3)$ are the vertices of a triangle, then its area is given by $\pm \frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

Given vertices are $(2,5),(7,k),(3,1)$ and area is $10$.

Therefore, $\pm 10=\frac{1}{2}\left[2\right(k-1)+7(1-5)+3(5-k\left)\right]$

$\Rightarrow \pm 20=2k-2-28+15-3k$

$\Rightarrow \pm 20=-k-15$
$\Rightarrow k=5$ or $-35$