Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.

Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.

Correct Answer: 1

Step 1: Identify the point from which we want to find the distance. The point is (1, 2).
Step 2: Write down the equation of the line in the form Ax + By + C = 0. The line is given as 3x + 4y - 12 = 0, so A = 3, B = 4, and C = -12.
Step 3: Use the distance formula for a point (x1, y1) to a line Ax + By + C = 0, which is Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2).
Step 4: Substitute the values into the formula. Here, x1 = 1, y1 = 2, A = 3, B = 4, and C = -12.
Step 5: Calculate the numerator: |3(1) + 4(2) - 12| = |3 + 8 - 12| = |3 - 12| = |1| = 1.
Step 6: Calculate the denominator: sqrt(A^2 + B^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Step 7: Divide the numerator by the denominator to find the distance: Distance = 1 / 5 = 1.

Distance from a Point to a Line – This concept involves using the formula for calculating the perpendicular distance from a point to a line represented in the standard form Ax + By + C = 0.
Understanding Line Equations – Recognizing the coefficients A, B, and C in the line equation and how they relate to the distance formula.
Absolute Value and Square Roots – Applying absolute value to ensure the distance is non-negative and using square roots to calculate the denominator.