For each question below, enter your answer as a decimal rounded to one decimal place. Find the length of the line segment shown below.

drawing two auxiliary line segments to create a right triangle whose hypotenuse is the original line segment
applying the Pythagorean Theorem

First, we can draw a vertical line segment from:

(1, -2) to (1, 4)

and a horizontal line segment from:

(1, 4) to (4, 4)

We can identify the lengths of these auxiliary line segments as:

6
3

See the attached image for a picture of these drawn on the graph.

Next, we can solve for the length of the hypotenuse, which is the original line segment, by plugging the lengths of the auxiliary line segments into the Pythagorean Theorem:

a² + b² = c²

where:

a and b are the lengths of the two shorter sides of a right triangle
c is the right triangle's hypotenuse (longest/diagonal) side

↓↓↓

6² + 3² = c²

36 + 9 = c²

45 = c²

c = √45

c ≈ 6.7

So, the line segment is about 6.7 units long.