Line ℓ has equation y=x–4. Find the distance between ℓ and the point T(1, - 6). Round your answer to the nearest tenth.

Answer 1

I assume the problem is to find the "shortest distance" between the point and the line!

There are 2 straightforward methods to solve. Let's use the perpendicular line method!

The line perpendicular to the line that passes through point (1,-6) is y_p = -x - 5

Where the 2 lines meet is the shortest distance, so let x-4 = -x-5, Solving for x, we get

x = -1/2, now solving for y in either equation gives, y = -9/2

So, the shortest distance is D = sqrt{ (-6+9/2)² + (1+1/2)² } = 2.12

Answer 2

The distance from a point to a line would be the length of the perpendicular from the point to the line.

The line y = x - 4 has slope 1, so a perpendicular line must have slope -1. The line through T(1, -6) with slope -1 has equation y - -6 = -1(x - 1), or y = -x - 5,

The intersection of the two lines y = x - 4 and y = -x - 5 is at (-1/2, -9/2).

The distance between T(1, -6) and (-1/2, -9/2) is √ [ (1 - -1/2)² + (-6 - -9/2)² ] ≈ 2.1.