Are the points $A(3, 6, 9)$, $B(10, 20, 30)$ and $C(25, -41, 5)$, the vertices of a right angled triangle?

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> [!TLDR]
> We check whether the three points form a right triangle by computing the squared distances between each pair and testing if the Pythagorean relation holds. The distances are $AB = 7\sqrt{14}$, $BC = \sqrt{6686}$, and $CA = \sqrt{3101}$; none of the squared sums match, so the triangle is **not** right-angled.

The question is: do three given points in 3D space form a right-angled triangle? In coordinate geometry, the cleanest way to test this is to compute the side lengths (using the distance formula) and then check if the square of the longest side equals the sum of the squares of the other two. That is the Pythagorean theorem in reverse — if it holds, the triangle is right-angled; if not, it isn’t.

We don’t need to find angles or slopes. Just distances. And because we’re in three dimensions, the distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

We’ll compute $AB$, $BC$, and $CA$, then compare squares.

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1. **Distance $AB$** between $A(3,6,9)$ and $B(10,20,30)$:

$$AB^2 = (10-3)^2 + (20-6)^2 + (30-9)^2 = 7^2 + 14^2 + 21^2$$
   $$= 49 + 196 + 441 = 686$$
   So $AB = \sqrt{686} = \sqrt{49 	imes 14} = 7\sqrt{14}$.

2. **Distance $BC$** between $B(10,20,30)$ and $C(25,-41,5)$:

$$BC^2 = (25-10)^2 + (-41-20)^2 + (5-30)^2 = 15^2 + (-61)^2 + (-25)^2$$
   $$= 225 + 3721 + 625 = 4571$$
   So $BC = \sqrt{4571}$.

3. **Distance $CA$** between $C(25,-41,5)$ and $A(3,6,9)$:

$$CA^2 = (3-25)^2 + (6-(-41))^2 + (9-5)^2 = (-22)^2 + 47^2 + 4^2$$
   $$= 484 + 2209 + 16 = 2709$$
   So $CA = \sqrt{2709}$. …

Quadrant	x-sign	y-sign	Example
I	$+$	$+$	$(2, 3)$
II	$-$	$+$	$(-1, 4)$
III	$-$	$-$	$(-3, -2)$
IV	$+$	$-$	$(5, -1)$

Q.Are the points $A(3, 6, 9)$ , $B(10, 20, 30)$ and $C(25, -41, 5)$ , the vertices of a right angled triangle?

Concept understanding — Coordinate Geometry

Coordinate Geometry: Where Algebra Meets Geometry

The Big Idea

The Coordinate System

A Concrete Example

The Four Quadrants

Why This Matters

The Precise Statement

Concept and Intuition

Step-by-Step Solution

Q.Are the points A(3,6,9)A(3, 6, 9)A(3,6,9), B(10,20,30)B(10, 20, 30)B(10,20,30) and C(25,−41,5)C(25, -41, 5)C(25,−41,5), the vertices of a right angled triangle?

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Q.Are the points $A(3, 6, 9)$ , $B(10, 20, 30)$ and $C(25, -41, 5)$ , the vertices of a right angled triangle?

Unlock everything free for 14 days