Show that the triangle ABC with vertices A(0,4,1), B(2,3,-1) and C(4,5,0) is right angled.

NCERT Exemplar · Q10

A(0,4,1)

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Concept understanding — Coordinate Geometry

Coordinate Geometry: Where Algebra Meets Geometry

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 $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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 …

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Using vectors, we show that two sides of triangle ABC are perpendicular by checking their dot product equals zero. The triangle is right-angled at vertex B.

Why vectors make this easy

In coordinate geometry, a right angle means two sides meet at 90°. With coordinates in 3D, the cleanest way to check perpendicularity is through vectors: if the dot product of two side vectors is zero, the angle between them is 90°.

We don't need to compute lengths or use the Pythagorean theorem — just form vectors from the given points and test their dot products.

Step-by-step solution

1. Choose a vertex to test.

Any vertex could be the right angle. We'll test vertex B first — if it's not right-angled there, we test another. But as we'll see, B works.

2. Form the two side vectors meeting at B.

From B to A:

$\overrightarrow{BA} = A - B = (0-2,\;4-3,\;1-(-1)) = (-2,\;1,\;2)$

From B to C:

$\overrightarrow{BC} = C - B = (4-2,\;5-3,\;0-(-1)) = (2,\;2,\;1)$

3. Compute the dot product.

$\overrightarrow{BA} \cdot \overrightarrow{BC} = (-2)(2) + (1)(2) + (2)(1) = -4 + 2 + 2 = 0$

Since the dot product is zero, the vectors are perpendicular. Therefore angle ABC = 90°.

Tip

You only need to check one pair of sides. If the dot product is zero, you're done — the triangle is right-angled at that vertex. No need to check the other two angles.

4. Confirm it's indeed a triangle (non-degenerate). …

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Q.Show that the triangle ABC with vertices A(0,4,1)A(0,4,1)A(0,4,1), B(2,3,−1)B(2,3,-1)B(2,3,−1) and C(4,5,0)C(4,5,0)C(4,5,0) is right angled.

Why vectors make this easy

Step-by-step solution

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Q.Show that the triangle ABC with vertices $A(0,4,1)$ , $B(2,3,-1)$ and $C(4,5,0)$ is right angled.

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