Find the distance from point A(-1/4,5) to the line -x+2y=14. Round your answer to the nearest tenth.

The distance from A(-1/4, 5) to the line -x + 2·y = 14 is approximately 1.7 units

Step-by-step explanation:

The shortest distance from a point to a line is the perpendicular from the line to the point

The equation of the given line is -x + 2·y = 14, which can be written in slope and intercept form as follows;

-x + 2·y = 14

2·y = 14 + x

y = 14/2 + x/2 = x/2 + 7

y = x/2 + 7

The slope, m₁ = 1/2 and the y-intercept = 7

Therefore, the slope, m₂, of the perpendicular line to the line -x + 2·y = 14 which was rewritten as y = x/2 + 7 is m₂ = -1/m₁ = -1/(1/2) = -2

Therefore, we have that the slope of the line from A(-1/4, 5) to the line -x + 2·y = 14, is -2

The equation of the line is therefore;

y - 5 = -2(x - (-1/4))

y - 5 = -2·x - 1/2

y = -2·x - 1/2 + 5 = -2·x + 9/2

y = -2·x + 9/2

The coordinates of the point on the line -x + 2·y = 14 (y = x/2 + 7) that coincides with the perpendicular from the point A(-1/4, 5) is therefore given as follows;

-2·x + 9/2 = x/2 + 7

9/2 - 7= x/2 + 2·x

-2.5 = 2.5·x

x = -2.5/2.5 = -1

x = -1

y = x/2 + 7 = -1/2 + 7 = 6.5

y = 6.5

The coordinates of the point on the line -x + 2·y = 14 (y = x/2 + 7) that coincides with the perpendicular from the point A(-1/4, 5) is (-1, 6.5)

The distance from A(-1/4, 5) to B(-1, 6.5), is given as follows;

Where;

(x₁, y₁) = (-1/4, 5), and (x₂, y₂) = B(-1, 6.5)

Plugging in the values, we have;

Therefore, the distance from A(-1/4, 5) to the line -x + 2·y = 14 ≈ 1.7 units, rounding to the nearest tenth.

step-by-step explanation:

fist on the left goes with the third on the right. second on the left goes with the fourth on the right. third on the left goes with the second on the right. the fourth on the left goes with first on the right