Question

The vertices of a triangle are $A\left(m,n\right),B\left(12,19\right)andC\left(23,20\right).$ its area is 70 square unit. If the slope of the median through A is -5,then $\left(m+n\right)$ is.

• A. 17
• B. 48
• C. 57
• D. 32

Answer 1

The correct option is D 32

Area of the triangle when vertices are given is $\frac{1}{2}|(x_A-x_C)(y_B-y_A)-(x_A-x_B)(y_C-y_A)|$

so, $\frac{1}{2}|(m-23)(19-n)-(m-12)(20-n)|=70\Rightarrow 19m-mn-437+23n-20m+mn+240-12n=140\Rightarrow 11n-m=337......\left(equation1\right)$

again slope of median through $A$ is $-5$

so, $\frac{n-19.5}{m-17.5}=-5\Rightarrow n+5m=107....\left(equation2\right)$

solving $\left(equation1\right)and\left(equation2\right)$ we get;

$m=15$ and $n=32$

so, $m+n=32$

so option D is correct