Question

e) The foot of a ladder is 9 m away from its wall, and its top reaches a window 12 m
above the ground.
(i) Find the length of the ladder.
(ii) If the ladder is shifted in a way that its foot stays 12 m away from the wall, to
what height does its top reach?​

Answer 1

We have :-

• the foot of ladder is 9 m away from its wall it means the distance b/w foot of ladder and wall i.e; base = 9 m
• the ladder top reaches a window 12 m away the ground that's mean height of window = 12 m

we have to find :-

1. Find the length of the ladder.
2. height of window if the distance of ladder's foot increase to 12 m .

Solution :-

1st case :-

• this case makes a right angle traingle .

so ,

• we have height = 12 m
• base = 9 m

the length of ladder = hypotenuse = ??

$\sf \: hypotenuse = \sqrt{ {(height)}^{2} + {(base)}^{2} } \\ \\ \sf \: hypotenuse = \sqrt{ {(12)}^{2} + {(9)}^{2} } \\ \\ \sf \: hypotenuse = \sqrt{225} \\ \\ \fbox{ \sf \: hypotenuse = 15 \: m \: }$

so length of ladder = 15 m

2nd case :-

• if the ladder is shifted in a way that its foot stays 12 m away from the wall, to
• what height does its top reach?

Solution :-

• now in this case base = 12m
• and we have find out the length of ladder that is 15 m
• here, we have to find out that to what height does ladder's top reach with the base 12 m .

$\sf \: {(hypotenuse}^{2} \: = {(height)}^{2} + {(base)}^{2} \\ \\ \sf \: {15}^{2} = {(height)}^{2} + {(12)}^{2} \\ \\ \sf \: 225= {(height)}^{2} + 144 \\ \\ \sf{(height)}^{2} = 225 - 144 \\ \\ \sf \: height = \sqrt{81} \\ \\ \fbox{ \sf \: height \: = 9 \: m}$

so new height = 9 m

Answer 2

Given :-

The foot of a ladder is 9 m away from its wall, and its top reaches a window 12 m  above the ground.

To Find :-

The length of the ladder.

If the ladder is shifted in a way that its foot stays 12 m away from the wall, find the height required to reach it's top.

Analysis :-

Firstly we have to find the length (hypotenuse) by it's respective formula.

Then find the height by substituting the values in the respective formula.

Solution :-

We know that,

• l = Length
• h = Height
• b = Base

By the formula,

$\underline{\boxed{\sf Hypotenuse=\sqrt{(Height)^2+(Base)^2} }}$

Given that,

Height (h) = 12 m

Base (b) = 9 m

Substituting their values,

$\sf Hypotenuse=\sqrt{(12)^2+(9)^2}$

$\sf =\sqrt{225}$

$\sf = 15 \ m$

Therefore, the length of the ladder is 15 m.

By the formula,

$\underline{\boxed{\sf Hypotenuse^2=(Height)^2+(Base)^2}}$

Given that,

Length (l) = 15 m

Base (b) = 12 m

Substituting their values,

$\sf 15^2=(Height)^2=(12)^2$

$\sf 225=(Height)^2+144$

By transposing,

$\sf (Height)^2=225-144$

$\sf (Height)^2=81$

$\sf Height=\sqrt{81} =9 \ m$

Therefore, the height to reach the top is 9 m.