The distance between the points M (k, 7) and N (9, k) is $\sqrt{20}$ units. Given that k > 10, find the value of k. Answer k = ...

1. Apply the distance formula to the two given points.

For $M(k,7)$ and $N(9,k)$,
$MN=\sqrt{(9-k)^2+(k-7)^2}.$
Given $MN=\sqrt{20}$,
$\sqrt{(9-k)^2+(k-7)^2}=\sqrt{20}.$

2. Square both sides and simplify to obtain a quadratic equation.

$(9-k)^2+(k-7)^2=20$
$k^2-18k+81+k^2-14k+49=20$
$2k^2-32k+110=0$
$k^2-16k+55=0$

3. Solve the quadratic equation and select the value satisfying $k>10$.

$k^2-16k+55=0$
$(k-5)(k-11)=0$
$k=5 \text{ or } k=11.$
Since $k>10$,
$k=11.$