Find the distance between the points A(-1, 1) and B(1, 6), and the distance between C(5, 0) and D(4, 2), then say which segment is longer.

Step-by-step solution

1. 1

   Write the distance formula and note why signs do not matter. For points $(x_1,y_1)$ and $(x_2,y_2)$,

   $d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}.$

   Both differences are squared, so subtracting in the other order gives the same distance — a useful reassurance when one coordinate is negative.

2. 2

   Compute the differences for A and B.

   $x_2-x_1=1-(-1)=2,\qquad y_2-y_1=6-1=5.$

   The double negative is the one place a sign error creeps in: $1-(-1)$ is $2$, not $0$.

3. 3

   Find AB.

   $AB=\sqrt{2^{2}+5^{2}}=\sqrt{4+25}=\sqrt{29}\approx 5.385.$

   $29$ is prime, so the radical cannot be simplified.

4. 4

   Compute the differences for C and D, then find CD.

   $x_2-x_1=4-5=-1,\qquad y_2-y_1=2-0=2,$
   $CD=\sqrt{(-1)^{2}+2^{2}}=\sqrt{1+4}=\sqrt{5}\approx 2.236.$

   The negative difference squares away, as promised.

5. 5

   Compare. $\sqrt{29}>\sqrt{5}$ because $29>5$ and the square root is increasing, so $AB$ is the longer segment — roughly $2.4$ times as long.

6. 6

   Check with the Pythagorean picture. $AB$ is the hypotenuse of a right triangle with legs $2$ and $5$; $CD$ is the hypotenuse of one with legs $1$ and $2$. The distance formula is nothing more than the Pythagorean theorem applied to those legs.