Q.Using the distance formula, examine whether the points $(1, 1)$, $(2, 3)$ and $(4, 7)$ lie on a straight line.

Compute the three pairwise distances and check whether the distance between the two outer points equals the sum of the distances to the point in between.

Let $A(1,1)$, $B(2,3)$, $C(4,7)$. Using $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

$AB$: $\sqrt{(2-1)^2+(3-1)^2}=\sqrt{1+4}=\sqrt5$

$BC$: $\sqrt{(4-2)^2+(7-3)^2}=\sqrt{4+16}=\sqrt{20}=2\sqrt5$

$AC$: $\sqrt{(4-1)^2+(7-1)^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt5$

Checking whether $AB+BC=AC$:

$AB+BC=\sqrt5+2\sqrt5=3\sqrt5=AC$ …

An alternative check is to compute the slope from $A$ to $B$ and from $B$ to $C$ separately; both equal $2$ here, confirming the same …

Students sometimes compute all three distances correctly but then compare the wrong pair — checking $AB+AC$ against $BC$ instead of correctly testing $AB+BC$ against $AC$, since $ …