15.2 Distance, Midpoint & Parallel Lines

15.2 Distance, Midpoint & Parallel Lines

Quick Answer: Distance between two points is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Parallel lines have the same slope. On the PERT coordinate-plane competency, these formulas measure length, locate segment centers, and confirm geometric relationships without a drawn grid—as long as you square carefully and simplify radicals only when needed for the options.

Section 15.1 focused on slope and line equations. This section adds two pure geometry formulas (distance and midpoint) and deepens parallel-line recognition from equations. Many placement items mix skills: find midpoint, then slope of a side, then decide if two sides are parallel.

Distance Formula

Plot two points and connect them with a right triangle whose legs are horizontal and vertical. The legs have lengths |x₂ − x₁| and |y₂ − y₁|. By the Pythagorean theorem:

d² = (x₂ − x₁)² + (y₂ − y₁)² → d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Order of points does not matter. Squaring removes the effect of a negative difference, so you do not need absolute values inside the squares.

Worked Example 1 — Distance between two points

Find the distance between (1, 2) and (4, 6).

Step 1: Δx = 4 − 1 = 3, Δy = 6 − 2 = 4.

Step 2: d = √(3² + 4²) = √(9 + 16) = √25 = 5.

Answer: 5. This is the classic 3-4-5 right triangle on the plane.

Worked Example 2 — Distance with negatives

Distance between (−3, 5) and (2, −7).

Δx = 2 − (−3) = 5, Δy = −7 − 5 = −12.

d = √(5² + (−12)²) = √(25 + 144) = √169 = 13.

Answer: 13 (5-12-13 triple). Sign errors on Δy do not change the square, but forgetting to square both components does.

Worked Example 3 — Leave as simplified radical

Distance between (0, 0) and (2, 4).

d = √(2² + 4²) = √(4 + 16) = √20 = √(4 · 5) = 2√5.

If options show √20 and 2√5, both are equal; prefer the simplified form when both appear unless the question asks only for an exact match to √20.

Worked Example 4 — Horizontal and vertical shortcuts

• From (−2, 7) to (5, 7): only x changes → d = |5 − (−2)| = 7.
• From (3, −1) to (3, 8): only y changes → d = |8 − (−1)| = 9.

Using the full formula still works: √(0 + 9²) = 9. Use the shortcut to save time and reduce arithmetic errors.

Midpoint Formula

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) is the average of the endpoints:

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Think “halfway in x, halfway in y.” Midpoint is a point, not a length—report an ordered pair.

Worked Example 5 — Midpoint basics

Midpoint of (2, 8) and (6, 2):

M_x = (2 + 6)/2 = 4, M_y = (8 + 2)/2 = 5 → M = (4, 5).

Worked Example 6 — Midpoint with negatives and odds

Midpoint of (−3, 1) and (5, −4):

M_x = (−3 + 5)/2 = 2/2 = 1, M_y = (1 + (−4))/2 = (−3)/2 = −3/2 → M = (1, −3/2).

Fractions in midpoints are normal when sums are odd. Do not round to integers unless the options force a decimal form that matches.

Worked Example 7 — Endpoint from midpoint (reverse use)

M = (3, 4) is the midpoint of A(1, 2) and B(x, y). Find B.

(1 + x)/2 = 3 → 1 + x = 6 → x = 5. (2 + y)/2 = 4 → 2 + y = 8 → y = 6.

B = (5, 6).

Check: midpoint of (1, 2) and (5, 6) is ((1+5)/2, (2+6)/2) = (3, 4). ✓

Parallel Lines from Slopes and Equations

Two distinct non-vertical lines are parallel if and only if they have the same slope. Vertical lines (x = constant) are parallel to each other. Horizontal lines (y = constant) are parallel to each other.

Worked Example 8 — Compare after rewriting

Are 4x − 2y = 6 and y = 2x + 9 parallel?

Solve first for y: −2y = −4x + 6 → y = 2x − 3. Slope is 2. Second line already has slope 2. Different intercepts (−3 vs 9) → parallel distinct lines.

Worked Example 9 — Not parallel: look-alike coefficients

Compare 2x + 3y = 6 and 4x + 6y = 5.

First: 3y = −2x + 6 → y = −(2/3)x + 2, m = −2/3. Second: 6y = −4x + 5 → y = −(4/6)x + 5/6 = −(2/3)x + 5/6, m = −2/3.

Slopes match → parallel. (If the second constant made the equations proportional including C, they could be the same line; here 6/5 vs other ratios show they are distinct.)

Contrast with 2x + 3y = 6 and 2x − 3y = 1: second slope is 2/3, not equal → not parallel.

Worked Example 10 — Parallel line through a point (application)

Write the equation of the line through (4, −1) parallel to x + 2y = 8.

Rewrite: 2y = −x + 8 → y = −(1/2)x + 4, so m = −1/2.

Point-slope: y + 1 = −(1/2)(x − 4) → y + 1 = −(1/2)x + 2 → y = −(1/2)x + 1.

In standard form if needed: x + 2y = 2.

Applications on the Coordinate Plane

Worked Example 11 — Is a triangle isosceles?

Vertices A(0, 0), B(6, 0), C(3, 4). Compare side lengths.

AB = |6 − 0| = 6
AC = √(3² + 4²) = 5
BC = √((3−6)² + (4−0)²) = √(9 + 16) = 5

AC = BC = 5 ≠ AB → isosceles with AB as the base. Midpoint of AB is (3, 0); the median from C to that midpoint is vertical (same x = 3), a useful check on geometry word problems.

Worked Example 12 — Segment length and midpoint together

Endpoints P(−2, 3) and Q(4, −5).

Midpoint: ((−2+4)/2, (3+(−5))/2) = (1, −1).

Distance: Δx = 6, Δy = −8 → d = √(36 + 64) = √100 = 10.

Half the segment length is 5; the midpoint is halfway along the segment, not “half the coordinates” without averaging carefully when signs differ.

Worked Example 13 — Parallel sides of a quadrilateral (slope test)

Points: A(0, 0), B(4, 2), C(5, 5), D(1, 3). Check whether AB ∥ DC.

m_AB = (2 − 0)/(4 − 0) = 2/4 = 1/2
m_DC = (3 − 5)/(1 − 5) = (−2)/(−4) = 1/2

Equal slopes → AB ∥ DC. (You could likewise test AD and BC for a parallelogram.)

PERT Strategy Notes

1. Sketch lightly on scratch paper—labels reduce swapped coordinates.
2. For distance, compute (Δx)² and (Δy)² on separate lines before adding under the radical.
3. For midpoint, add then divide by 2—do not subtract (subtraction is for slope/distance deltas).
4. For parallel checks, always convert to y = mx + b (or spot both vertical) before comparing.
5. When options include both simplified and unsimplified radicals, compute fully; match the form shown.

Distance, midpoint, and parallel slope tests are mechanical once memorized. They reward careful arithmetic more than clever tricks—exactly the kind of items that raise a Math placement score when you slow down for one clean substitution pass.