Applying Slope Criteria to Lines

Lesson 2 of 2: Using the Criteria

In this lesson:

• Write equations of parallel and perpendicular lines through a point
• Apply slope criteria to geometric problems

Learning Objectives

By the end of this lesson, you should be able to:

3. Find the equation of a line parallel to a given line through a specified point
4. Find the equation of a line perpendicular to a given line through a specified point
5. Apply slope criteria to solve geometric problems involving parallel and perpendicular lines

Criteria Recap Before Applying Them

From Lesson 1 — two criteria, both proven:

• Parallel lines: same slope →
• Perpendicular lines: product = →

Tool for writing equations: point-slope form

Three-Step Procedure for Both Types

Step 1: Identify the slope of the given line (rewrite in slope-intercept form if needed)

Step 2: Determine the new slope:

• Parallel → same slope
• Perpendicular → negative reciprocal

Step 3: Use point-slope form with the new slope and given point

Example: Parallel Line Through a Point

Find the line through parallel to .

• Given slope:
• Parallel slope: (same)
• Point-slope form:
• Simplify:

Example: Parallel Line From Standard Form

Find the line through parallel to .

Step 1: Rewrite: . Slope

Step 2: Parallel slope

Step 3:

Your Turn: Write a Parallel Line

Find the line through parallel to .

Apply the three-step procedure — try it before the next slide.

Answer: Parallel Line Through (1, 0)

Step 1: Slope of given line

Step 2: Parallel slope

Step 3:

Example: Perpendicular Line Through a Point

Find the line through perpendicular to .

• Given slope:
• Perpendicular slope:
• Point-slope:
• Simplify:

Common Error: Using the Given Slope

Common error: after identifying slope , using that same slope in point-slope form gives a parallel line, not a perpendicular one.

Write both slopes explicitly before using point-slope form:

• Given slope:
• New slope: ← this is what goes into point-slope

Your Turn: Write a Perpendicular Line

Find the line through perpendicular to .

Work all three steps — write the new slope explicitly before using point-slope.

Answer: Perpendicular Line Through (1, 0)

Step 1: Given slope

Step 2: Perpendicular slope

Step 3:

Product check: ✓

Slope Criteria as Geometric Tools

The criteria aren't just for writing equations — they solve geometric problems:

• Altitude of a triangle: perpendicular from a vertex to the opposite side
• Perpendicular bisector: perpendicular to a segment at its midpoint
• Verify a right angle: check if slope product =

Example: Altitude of a Triangle

Triangle : , ,

Find the altitude from to side .

Step 1: Slope of

Step 2: Altitude slope (negative reciprocal)

Step 3: Altitude through : →

Example: Perpendicular Bisector of a Segment

Midpoint ; slope → bisector slope → equation:

Example: Verifying a Right Angle

Show that triangle with vertices , , has a right angle.

• Slope from to :
• Slope from to :

Product: ✓

Right angle at vertex .

Quick Check: Altitude From a Vertex

Triangle: , ,

Find the equation of the altitude from to side .

Identify slope of BC, find the perpendicular slope, write the altitude equation.

Answer: Altitude From Vertex A

Slope of :

Altitude slope (negative reciprocal):

Altitude through :

Product check: ✓

Key Takeaways

✓ Parallel: same slope in point-slope form

✓ Perpendicular: negative reciprocal; product

✓ Altitude: perpendicular from vertex to opposite side

✓ Perp bisector: perpendicular through midpoint of segment

Use the new slope, not the given slope

Slopes and : product , not perpendicular

Always verify:

What Comes Next

This lesson connects to:

• HSG.GPE.B.4 — Coordinate proofs using parallel/perpendicular slope criteria
• HSG.GPE.B.7 — Computing distances using perpendicular segments
• Calculus — Tangent and normal lines to curves use this exact perpendicular slope relationship

You now have: proven slope criteria + tools to build altitudes, bisectors, and right-angle tests — the foundation for coordinate geometry.