What is the perpendicular distance from the point 2 4 to the straight line y equals 2x plus 10?

Answer 1

It is 2√5 ≈ 4.47 units.

To solve this:

1. Find the equation of the line perpendicular to y = 2x + 10 that passes through the point (2, 4);
2. Find the point where this line meets y = 2x + 10
3. Find the distance from this point to (2, 4) using Pythagoras

The slope of the perpendicular line (m') to a line with slope m is such that mm' = -1, ie m' = -1/m

For y = 2x + 10, the perpendicular line has slope -1/2, and so the line that passes through (2, 4) with this slope is given by:

y - 4 = -½(x - 2)

→ 2y - 8 = -x + 2

→ 2y + x = 10

To find where this meets the line y = 2x + 10, substitute for y in the equation of the perpendicular line and solve for x:

y = 2x + 10

2y + x = 10

→ 2(2x + 10) + x = 10

→ 4x + 20 + x = 10

→ 5x = -10

→ x = -2

Now use one of the equations to solve for y:

y = 2x + 10

→ y = 2(-2) + 10

→ y = -4 + 10 = 6

This the perpendicular line from (2, 4) meets the line y = 2x + 10 at the point (-2, 6)

The distance between these two points is given by:

distance = √((-2 - 2)² + (6 - 4)²) = √(16 + 4) = √20 = 2√5 ≈ 4.47 units