Before any formula, think about what a straight line does. If you walk from your home to school, the shortest path is a straight line. If you stretch a thread tight between two pins, it forms a straight line. The key property is no turning — the direction never changes.
In coordinate geometry, we describe this mathematically. A straight line is the set of all points (x,y) that satisfy a simple rule: the slope (steepness) between any two points on the line is constant.
The Precise Definition
A straight line in the xy-plane is the graph of a linear equation in two variables. The most general form is:
Ax+By+C=0
where A, B, and C are real numbers, and A and B are not both zero.
Important
Every straight line corresponds to a linear equation, and every linear equation (with x and y) graphs as a straight line. This is the fundamental link between algebra and geometry.
The Slope: Why It Matters
The slopem measures how steep the line is. It is the ratio of the vertical change to the horizontal change between any two points (x1,y1) and (x2,y2) on the line:
m=x2−x1y2−y1
If the line is horizontal, m=0. If it is vertical, the slope is undefined (division by zero). A positive slope means the line rises as you move right; a negative slope means it falls.
Tip
For a vertical line, the equation is simply x=constant. For a horizontal line, it is y=constant.
Common Forms of the Equation
You will encounter these forms frequently. Each is useful in different situations.
1. Slope-Intercept Form
y=mx+c
Here m is the slope and c is the y-intercept — the point where the line crosses the y-axis. This is the most intuitive form: you see the slope and the starting height immediately.
2. Point-Slope Form
y−y1=m(x−x1)
Use this when you know the slope m and one point (x1,y1) on the line.
3. Two-Point Form
y2−y1y−y1=x2−x1x−x1
Use this when you know two distinct points on the line.
4. Intercept Form
ax+by=1
Here a is the x-intercept and b is the y-intercept. This form is convenient when the line cuts the axes at known points.
Ax+By+C=0
This is the general form. All other forms can be rearranged into it.
Special Cases to Watch For
Line through the origin: y=mx (no constant term).
Horizontal line: y=k (slope 0).
Vertical line: x=k (slope undefined).
Parallel lines: Two lines are parallel if they have the same slope (m1=m2).
Perpendicular lines: Two lines are perpendicular if the product of their slopes is −1 (m1⋅m2=−1), provided neither is vertical.
Watch out
The perpendicular condition m1m2=−1 fails when one line is vertical (undefined slope). In that case, the other line must be horizontal (m=0). Memorise this exception.
Distance Between a Point and a Line
Given a line Ax+By+C=0 and a point (x1,y1), the perpendicular distance d is:
d=A2+B2∣Ax1+By1+C∣
This formula is exact and works for any line. The absolute value ensures distance is always positive.
Putting It Together
When you see a problem about a straight line, ask yourself:
What information do I have? (slope, one point, two points, intercepts)
Which form of the equation fits best?
Do I need to find slope, intercepts, distance, or intersection with another line?
The straight line is the simplest curve in coordinate geometry, but it is the foundation for everything that follows — circles, parabolas, and beyond. Master it well.
Each requested line uses the midpoint and slope of side BC: the median passes through A and the midpoint of BC, the altitude passes through A perpendicular to BC, and the perpendicular bisector passes through the midpoint of BC perpendicular to it.
✓Final answer
Median through A: 13x−y−9=0.
Altitude through A: 3x−y+1=0.
Perpendicular bisector of BC: 3x−y−4=0.
Find the midpoint and slope of BC, then build each requested line using the point-slope form.
Preliminary: midpoint D of BC (needed for both the median and the perpendicular bisector):
D=(22+(−1),2−3+(−2))=(21,−25)
Preliminary: slope of BC (needed for both the altitude and the perpendicular bisector):
mBC=−1−2−2−(−3)=−31=−31
(i) Median through A — the line joining A(1,4) to D(21,−25).
mAD=21−1−25−4=−21−213=13
y−4=13(x−1)⟹y=13x−13+4=13x−9⟹13x−y−9=0
(ii) Altitude through A — perpendicular to BC, passing through A(1,4).
m⊥=−mBC1=−−1/31=3
y−4=3(x−1)⟹y=3x−3+4=3x+1⟹3x−y+1=0
(iii) Perpendicular bisector of BC — perpendicular to BC (same slope 3 as above), through midpoint D(21,−25).
y−(−25)=3(x−21)⟹y+25=3x−23
y=3x−23−25=3x−4⟹3x−y−4=0
Self-check. Median passes through both A(1,4): 13−4−9=0✓, and D(21,−25): 13(21)−(−25)−9=6.5+2.5−9=0✓. Altitude passes through A: 3−4+1=0✓, and mBC×3=−31×3=−1✓ (perpendicular). Perpendicular bisector passes through D: 3(0.5)−(−2.5)−4=1.5+2.5−4=0✓.