Trapezoid $ABCD$ has vertices $A(-1,0)$, $B(0,4)$, $C(m,4)$ and $D(k,0)$, with $m>0$ and $k>0$. The line $y = -x + 4$ is perpendicular to the line containing side $CD$, and the area of trapezoid $ABCD$ is 34 square units. What is the value of $k$?
+128475 Since the line y = -x + 4 has a slope of -1, then the line containing CD has a negative reciprocal slope = 1
Therefore....the slope of this line is
(4 - 0) / ( m - k) = 1
4 = m - k
4 + k = m
Now...the height of the trapezoid is 4
And the length of one base is k - - 1 = k + 1
And the length of the other base is 4 + k
So....the area of the trapezoid can be represented as
(1/2) height [ length of base 1 + length of base 2 ] = 34
(1/2) 4 * [ ( k + 1) + ( 4 + k) ] =34
2 * [ 2k + 5 ] = 34
2k + 5 = 17
2k =12
k = 6
Here is a pic :
