NCERT Solutions for Class 9 Maths Chapter 13 — Exercise Set 13.3#
$$\text{Slope} = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$$A larger slope means a steeper line. A horizontal line has slope 0; a vertical line has no defined slope.
1. The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest. (A: rise 60 m, run 70 m; B: rise 60 m, run 110 m; C: rise 80 m, run 100 m) #
Solution
$$\text{Slope of A} = \dfrac{60}{70} \approx 0.86$$
$$\text{Slope of B} = \dfrac{60}{110} \approx 0.55$$
$$\text{Slope of C} = \dfrac{80}{100} = 0.80$$$$\Rightarrow 0.55 < 0.80 < 0.86$$
∴ From least to greatest steepness: B, C, A.
2. The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp. #
Solution
$$\text{Slope} = \dfrac{2.5}{4}$$
$$\text{Slope} = \dfrac{5}{8} = 0.625$$
3. Find the slope of the line l in each of the following diagrams. #
Each big square of the grid is 1 unit.
(i) Horizontal line
The line $l$ is horizontal, so it has no rise: any two points on it have the same $y$-coordinate.
$$\text{Slope} = \dfrac{0}{\text{run}} = 0$$(ii) Line through (−2, 2) and (2, −2)
$$\text{Slope} = \dfrac{-2 - 2}{2 - (-2)}$$
$$\text{Slope} = \dfrac{-4}{4} = -1$$
The line goes down from left to right, so its slope is negative.
(iii) Line through (−2, −1) and (1, 1)
$$\text{Slope} = \dfrac{1 - (-1)}{1 - (-2)}$$
$$\text{Slope} = \dfrac{2}{3}$$
(iv) Vertical line
The line $l$ is vertical: any two points on it have the same $x$-coordinate, so the run is 0. Division by 0 is not possible, so the slope of a vertical line is not defined.
4. An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm? #
Solution
The slope must be $\dfrac{1}{12}$. Let the horizontal length be $L$ cm. The rise is 18 cm, so:
$$\dfrac{18}{L} = \dfrac{1}{12}$$$$\Rightarrow L = 18 \times 12$$
$$\Rightarrow L = 216$$
∴ The ramp should be 216 cm (2.16 m) long horizontally.