LEARN · EXPLAIN · REVISE

Find the distance from P to the line.

Read the idea, work independently, then explain what changed.

高二選擇性必修第一册(A版).pdf · 2.3 · PDF 75 / printed page 70

TOPIC 01

Intersections and distance formulas

Solve intersections and calculate point-line and parallel-line distances with geometric checks.

Solve intersections and calculate point-line and parallel-line distances with geometric checks.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

P=(0,0),\quad 3x+4y-35=0

The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Use this intermediate relation.

d=|C|/5

Worked solution

The requested value is 7.

Checks and common pitfalls: The normal (3,4) has length 5.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Solve intersections and calculate point-line and parallel-line distances with geometric checks.
Which condition is essential in intersections and distance formulas?
Why does multiplying a line equation by 100 leave its distance formula unchanged?

Intersection: Solve both linear equations simultaneously and check both.
Normalized distance: The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
$d=|Ax_0+By_0+C|/\sqrt{A^2+B^2}$

Subtracting constants before matching parallel-line coefficients gives a wrong distance.