Q.The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.
Imagine you're telling a friend where you left your book in a library. You don't say "near the window" — you say "third shelf, second row, fourth book from the left." You're using numbers to pin down an exact location.
Coordinate geometry does the same thing, but for points on a flat surface. It gives every point a precise address — a pair of numbers — so we can describe shapes, distances, and positions using algebra.
The Big Idea
Before coordinate geometry, geometry was about drawing shapes and proving things with logic alone. Algebra was about numbers and equations. These two worlds seemed separate.
Then René Descartes (a French mathematician) had a simple but revolutionary idea: draw two perpendicular number lines that cross at zero. Now every point on the plane has a unique pair of numbers — its coordinates.
That's it. That's the entire foundation.
The Coordinate System
Take a horizontal line — call it the x-axis. Take a vertical line — call it the y-axis. They cross at a point called the origin, labelled O.
Any point P is located by two numbers:
Its x-coordinate: how far right (positive) or left (negative) from the origin
Its y-coordinate: how far up (positive) or down (negative) from the origin
We write this as an ordered pair: (x,y).
Note
The order matters. (3,5) is not the same point as (5,3). The first number is always the horizontal position; the second is always the vertical.
A Concrete Example
Plot the point A(2,3):
Start at the origin (0,0).
Move 2 units to the right along the x-axis.
From there, move 3 units up (parallel to the y-axis).
Mark the point.
Now plot B(−1,4):
Start at the origin.
Move 1 unit left (negative x-direction).
Move 4 units up.
Mark the point.
Every point on the plane has exactly one such address. And every pair of numbers corresponds to exactly one point. This one-to-one matching is what makes coordinate geometry powerful.
The Four Quadrants
The axes divide the plane into four regions, called quadrants:
Quadrant
x-sign
y-sign
Example
I
+
+
(2,3)
II
−
+
(−1,4)
III
−
−
(−3,−2)
IV
+
−
(5,−1)
Points on the axes themselves (where either coordinate is zero) don't belong to any quadrant.
Why This Matters
Once every point has a number address, we can:
Calculate distances between points using the Pythagorean theorem
Find midpoints by averaging coordinates
Describe lines with equations like y=mx+c
Solve geometric problems using algebra instead of drawing
Important
The distance between two points (x1,y1) and (x2,y2) is:
d=(x2−x1)2+(y2−y1)2
This is just the Pythagorean theorem in disguise.
The Precise Statement
Coordinate geometry (also called analytic geometry) is the study of geometry using a coordinate system. It establishes a correspondence between:
Points on a plane and ordered pairs of real numbers
Geometric figures (lines, circles, curves) and algebraic equations
This correspondence lets us translate geometric problems into algebraic ones, solve them with equations, and translate the answers back into geometric meaning.
A Simple Application
Find the distance between P(1,2) and Q(4,6).
Using the formula:
d=(4−1)2+(6−2)2=32+42=9+16=25=5
The distance is 5 units. You could verify this by plotting the points and drawing a right triangle — the horizontal leg is 3, the vertical leg is 4, and the hypotenuse is 5. The formula just automates that reasoning.
What Comes Next
Once you're comfortable with coordinates, you'll learn to:
Write equations of lines (y=mx+c)
Find slopes and intercepts
Work with circles (x2+y2=r2)
Solve problems involving midpoints, section formulas, and areas of triangles
But it all rests on this one idea: every point has a number address, and every number address points to exactly one location. That bridge between numbers and space is the heart of coordinate geometry.
Coordinate Geometry is one of the largest, most consistently weighted units across the NCERT Class 9 to 11 Mathematics curriculum, and it's exactly the topic behind searches like "coordinate geometry: definition, formula and examples" or "coordinate geometry important questions class 10". Mastering this foundational bridge between algebra and geometry pays off across CBSE boards, JEE Main, and virtually every state CET exam's geometry section.
Concept: Coordinate Geometry — placing a symmetric figure on axes to simplify coordinates.
Step 1 – Base on y-axis, midpoint at origin
Let the base vertices be B and C. Since the midpoint is at (0,0) and the base lies along the y-axis, the coordinates are symmetric:
B=(0,a) and C=(0,−a).
Step 2 – Third vertex lies on perpendicular bisector
The perpendicular bisector of the base is the x-axis (since base is vertical). So the third vertex A has coordinates (h,0).
Step 3 – Use side length to find h
Side length is 2a. Distance from A to B:
(h−0)2+(0−a)2=h2+a2=2a
Squaring: h2+a2=4a2⇒h2=3a2⇒h=±a3.
Thus the two possible positions for A are (a3,0) and (−a3,0).
✓Final answer
The vertices are (0,a), (0,−a), and (±a3,0).
The key idea is to place the base symmetrically about the origin on the y-axis, then use the height formula for an equilateral triangle to locate the third vertex on the x-axis. The vertices are (0,a), (0,−a), and (3a,0).
We are told the base of an equilateral triangle has length 2a, lies along the y-axis, and its midpoint is at the origin. That means the base is a vertical segment centered at (0,0). The third vertex will be somewhere on the perpendicular bisector of this base — which, because the base is vertical, is the x-axis. For an equilateral triangle, the altitude is a fixed multiple of the side length, so we can compute exactly where that third vertex lies.
Let’s work through it.
Place the base on the y-axis.
Since the midpoint is at (0,0) and the base length is 2a, the two endpoints of the base are at (0,a) and (0,−a). These are the two vertices on the y-axis.
Find the altitude of the equilateral triangle.
For any equilateral triangle of side s, the altitude is
h=23s.
Here s=2a, so
h=23⋅2a=3a.
Locate the third vertex.
The altitude from the base goes along the perpendicular bisector. The base is vertical, so its perpendicular bisector is horizontal — the x-axis. The third vertex is therefore on the x-axis, at a distance h from the base’s midpoint. That gives two possibilities: to the right or to the left.
So the third vertex is at (3a,0) or (−3a,0).
Tip
The problem doesn’t specify which side of the y-axis the triangle lies on, so both are valid. Usually we take the positive x-direction unless told otherwise.
Verify the distances.
Check that the distance from (0,a) to (3a,0) is indeed 2a:
(3a−0)2+(0−a)2=3a2+a2=4a2=2a.
The same holds for the other vertex. So the triangle is equilateral.
Watch out
A common mistake is to forget that the base is along the y-axis, not centered at the origin with its endpoints at (0,0) and (0,2a). The phrase “mid-point of the base is at the origin” forces the symmetric placement we used.
✓Final answer
The vertices are (0,a), (0,−a), and (3a,0) (or (−3a,0)).