The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensi

NCERT Exemplar · Q47

Q. The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are ______.

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Concept understanding — Collinearity Slope Condition

Collinearity Slope Condition — From Intuition to Precision

Imagine three points scattered on a sheet of paper. You can always draw a triangle through them. But if those three points happen to lie perfectly on a single straight line — like beads on a thread — they are called collinear (from Latin co- meaning "together" and linearis meaning "belonging to a line").

The question is: how do you check, using only coordinates, whether three given points are collinear?

The Intuition

Think about walking from point A to point B, then from point B to point C. If all three lie on the same line, your direction of travel should not change when you turn at B. In other words, the slope of AB must equal the slope of BC.

Slope measures steepness: $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ . If two segments share the same slope and meet at a common point (B), they lie on the same straight line.

Note

This works because a line has a constant slope everywhere. If AB and BC have the same slope, they are parts of the same line — they cannot bend.

The Precise Statement

Let three points be $A(x_1, y_1)$ , $B(x_2, y_2)$ , and $C(x_3, y_3)$ . They are collinear if and only if:

\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}

provided that $x_1 \neq x_2$ and $x_2 \neq x_3$ (i.e., no vertical segment).

\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}

What About Vertical Lines?

If $x_1 = x_2$ , the slope of AB is undefined (division by zero). But the condition still works: if AB is vertical, then for collinearity, BC must also be vertical — meaning $x_2 = x_3$ . So the condition becomes: either both slopes are equal and defined, or both are undefined (i.e., both segments are vertical).

A Cleaner Algebraic Form

Cross-multiplying the slope equality gives a form that avoids division entirely:

(y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1)

This works for all cases, including vertical lines.

Tip

For quick checks, use the cross-multiplied form — no need to worry about zero denominators.

Example

Check if $A(1, 2)$ , $B(3, 6)$ , $C(5, 10)$ are collinear.

Slope of AB: $\frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$

Slope of BC: $\frac{10 - 6}{5 - 3} = \frac{4}{2} = 2$

Slopes are equal → points are collinear. Indeed, they all lie on $y = 2x$ .

Why This Matters

The slope condition is the simplest test for collinearity in coordinate geometry. It appears in: …

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The longest piece of string that can be stretched straight in a rectangular room is its space diagonal, calculated using the 3D Pythagorean theorem. The length is $\boxed{3\sqrt{37}}$ units.

A rectangular room is a three-dimensional shape known as a cuboid. When we talk about stretching a string straight within this room, we are looking for the longest possible straight line segment that can fit entirely inside it.

Concept and Intuition: The Space Diagonal

Imagine you are standing in one corner of the room, say the bottom-front-left corner. If you want to stretch a string to the furthest possible point in a straight line, you would aim for the opposite corner – the top-back-right corner. This line segment, connecting two opposite vertices of the cuboid, is called the space diagonal. Any other line segment within the room (like an edge, a face diagonal, or a line connecting non-opposite points) will be shorter than or equal to this space diagonal.

The length of this space diagonal can be found by extending the familiar Pythagorean theorem into three dimensions. Just as a 2D diagonal in a rectangle with sides $a$ and $b$ is $\sqrt{a^2 + b^2}$ , a 3D diagonal in a cuboid with dimensions $l$ , $w$ , and $h$ is $\sqrt{l^2 + w^2 + h^2}$ .

D

Let's apply this concept to the given dimensions.

Identify the dimensions:

The dimensions of the rectangular room are given as 10, 13, and 8 units. We can assign these to the length ( $l$ ), width ( $w$ ), and height ( $h$ ) in any order, as addition is commutative.

Let $l = 10$ units, $w = 13$ units, and $h = 8$ units.
Apply the space diagonal formula: …

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Q.The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are ______.

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