A 10 m long ladder reaches a window 8 m high from a wall at a distance x. Find the distance 'x'.

Question

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2.

Find the length of AD using the Pythagorean theorem in $\triangle ABD$ .
In right-angled $\triangle ABD$ :
$AB^2 = AD^2 + BD^2$ $13^2 = AD^2 + 5^2$ $169 = AD^2 + 25$ $AD^2 = 169 - 25$ $AD^2 = 144$ $AD = \sqrt{144} = 12 \text{ cm}$
Find the length of $x$ (CD) using the Pythagorean theorem in $\triangle ADC$ .
In right-angled $\triangle ADC$ :
$AC^2 = AD^2 + CD^2$ $15^2 = 12^2 + x^2$ $225 = 144 + x^2$ $x^2 = 225 - 144$ $x^2 = 81$ $x = \sqrt{81} = 9 \text{ cm}$
The value of $x$ is $\boxed{\text{9 cm}}$ .

3.
Identify the components of the right-angled triangle.
The ladder, the wall, and the ground form a right-angled triangle.
Hypotenuse (ladder length) $= 10$ m
One leg (height of window) $= 8$ m
Other leg (distance from wall) $= x$ m
Apply the Pythagorean theorem.
$(\text{ladder length})^2 = (\text{height of window})^2 + (\text{distance from wall})^2$ $10^2 = 8^2 + x^2$ $100 = 64 + x^2$ $x^2 = 100 - 64$ $x^2 = 36$ $x = \sqrt{36} = 6 \text{ m}$
The distance $x$ is $\boxed{\text{6 m}}$ .

4.
A triplet $(a, b, c)$ is a Pythagorean triplet if $a^2 + b^2 = c^2$ , where $c$ is the largest number.
- i) 3, 7, 9

Check if $3^2 + 7^2 = 9^2$ .

$$
3^2 + 7^2 = 9 + 49 = 58
$$
$$
9^2 = 81
$$
Since $58 \neq 81$, (3, 7, 9) is **not** a Pythagorean triplet.

ii) 7, 24, 25

Check if $7^2 + 24^2 = 25^2$ .
$$ 7^2 + 24^2 = 49 + 576 = 625 $$ $$ 25^2 = 625 $$ Since $625 = 625$, (7, 24, 25) **is** a Pythagorean triplet.
The Pythagorean triplet is $\boxed{\text{(7, 24, 25)}}$ .

5.
Identify the known sides of the right-angled triangle.
Hypotenuse $= 25$ cm
One leg $= 15$ cm
Let the third side be $y$ cm.
Apply the Pythagorean theorem.
$(\text{hypotenuse})^2 = (\text{leg}_1)^2 + (\text{leg}_2)^2$ $25^2 = 15^2 + y^2$ $625 = 225 + y^2$ $y^2 = 625 - 225$ $y^2 = 400$ $y = \sqrt{400} = 20 \text{ cm}$
The length of its third side is $\boxed{\text{20 cm}}$ .

6.
Visualize the movement as a right-angled triangle.
The man's movement east (10 m) and north (24 m) form the two legs of a right-angled triangle. His displacement from the starting point is the hypotenuse.
Leg 1 (East) $= 10$ m
Leg 2 (North) $= 24$ m
Let the displacement be $d$ m.
Apply the Pythagorean theorem.
$d^2 = (\text{East})^2 + (\text{North})^2$ $d^2 = 10^2 + 24^2$ $d^2 = 100 + 576$ $d^2 = 676$ $d = \sqrt{676} = 26 \text{ m}$
The man's displacement from the starting point is $\boxed{\text{26 m}}$ .

7.
Identify the sides of the triangle and the longest side.
Sides are 15 cm, 36 cm, and 39 cm. The longest side is 39 cm.
For a triangle to be right-angled, the square of the longest side must be equal to the sum of the squares of the other two sides (Pythagorean theorem).
Check if the Pythagorean theorem holds.
$15^2 + 36^2 = 225 + 1296 = 1521$ $39^2 = 1521$
Since $15^2 + 36^2 = 39^2$ , the triangle is a right-angled triangle.
The triangle is a $\boxed{\text{right-angled triangle}}$ .

8.
Identify the components of the right-angled triangle.
The ladder, the wall, and the ground form a right-angled triangle.
Hypotenuse (ladder length) $= 4.5$ m
One leg (distance from wall) $= 2.7$ m
Other leg (height on wall) $= h$ m
Apply the Pythagorean theorem.
$(\text{ladder length})^2 = (\text{distance from wall})^2 + (\text{height on wall})^2$ $4.5^2 = 2.7^2 + h^2$ $20.25 = 7.29 + h^2$ $h^2 = 20.25 - 7.29$ $h^2 = 12.96$ $h = \sqrt{12.96} = 3.6 \text{ m}$
The height of the wall where the ladder reaches is $\boxed{\text{3.6 m}}$ .

9.
Visualize the setup and form a right-angled triangle.
Draw a horizontal line from the top of the shorter pole to the taller pole. This forms a right-angled triangle.
The horizontal distance between the poles is one leg $= 12$ m.
The difference in heights of the poles is the other leg $= 14 \text{ m} - 9 \text{ m} = 5$ m.
The distance between their tops is the hypotenuse. Let this be $d$ .
Apply the Pythagorean theorem.
$d^2 = (\text{horizontal distance})^2 + (\text{difference in heights})^2$ $d^2 = 12^2 + 5^2$ $d^2 = 144 + 25$ $d^2 = 169$ $d = \sqrt{169} = 13 \text{ m}$
The distance between their tops is $\boxed{\text{13 m}}$ .

10.
Identify the sides of the triangle and the longest side.
Sides are 6.5 cm, 6 cm, and 2.5 cm. The longest side is 6.5 cm.
For a triangle to be right-angled, the square of the longest side must be equal to the sum of the squares of the other two sides.
Check if the Pythagorean theorem holds.
$6^2 + 2.5^2 = 36 + 6.25 = 42.25$ $6.5^2 = 42.25$
Since $6^2 + 2.5^2 = 6.5^2$ , the triangle is a right-angled triangle.
If it is a right-angled triangle, the hypotenuse is always the longest side.
The hypotenuse is $\boxed{\text{6.5 cm}}$ .

11.
Identify the components of the right-angled triangle formed by the diagonal.
The adjacent sides of a rectangle form the legs of a right-angled triangle, and the diagonal is the hypotenuse.
Leg 1 $= 3.6$ cm
Leg 2 $= 1.5$ cm
Let the diagonal be $d$ cm.
Apply the Pythagorean theorem.
$d^2 = (\text{side}_1)^2 + (\text{side}_2)^2$ $d^2 = 3.6^2 + 1.5^2$ $d^2 = 12.96 + 2.25$ $d^2 = 15.21$ $d = \sqrt{15.21} = 3.9 \text{ cm}$
The length of its diagonal is $\boxed{\text{3.9 cm}}$ .

12.
Identify the properties of an isosceles right triangle.
An isosceles right triangle has two equal sides (legs) and a right angle. Since it's right-angled at C, the legs are AC and BC.
Given $AC = 4$ cm, then $BC = AC = 4$ cm.
AB is the hypotenuse.
Apply the Pythagorean theorem.
$AB^2 = AC^2 + BC^2$ $AB^2 = 4^2 + 4^2$ $AB^2 = 16 + 16$ $AB^2 = 32$ $AB = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \text{ cm}$
The length of AB is $\boxed{4\sqrt{2} \text{ cm}}$ .

A 10 m long ladder reaches a window 8 m high from a wall at a distance x. Find the distance 'x'.

Step-by-step solution

Find the length of AD using the Pythagorean theorem in $\triangle ABD$ .

Find the length of $x$ (CD) using the Pythagorean theorem in $\triangle ADC$ .

Identify the components of the right-angled triangle.

Apply the Pythagorean theorem.

Check if $3^2 + 7^2 = 9^2$ .

Check if $7^2 + 24^2 = 25^2$ .

Identify the known sides of the right-angled triangle.

Apply the Pythagorean theorem.

Visualize the movement as a right-angled triangle.

Apply the Pythagorean theorem.

Identify the sides of the triangle and the longest side.

Check if the Pythagorean theorem holds.

Identify the components of the right-angled triangle.

Apply the Pythagorean theorem.

Visualize the setup and form a right-angled triangle.

Apply the Pythagorean theorem.

Identify the sides of the triangle and the longest side.

Check if the Pythagorean theorem holds.

Identify the components of the right-angled triangle formed by the diagonal.

Apply the Pythagorean theorem.

Identify the properties of an isosceles right triangle.

Apply the Pythagorean theorem.

Still stuck on this one?

More Mathematics Questions

A 10 m long ladder reaches a window 8 m high from a wall at a distance x. Find the distance 'x'.

Step-by-step solution

Find the length of AD using the Pythagorean theorem in △ABD\triangle ABD△ABD.

Find the length of xxx (CD) using the Pythagorean theorem in △ADC\triangle ADC△ADC.

Identify the components of the right-angled triangle.

Apply the Pythagorean theorem.

Check if 32+72=923^2 + 7^2 = 9^232+72=92.

Check if 72+242=2527^2 + 24^2 = 25^272+242=252.

Identify the known sides of the right-angled triangle.

Apply the Pythagorean theorem.

Visualize the movement as a right-angled triangle.

Apply the Pythagorean theorem.

Identify the sides of the triangle and the longest side.

Check if the Pythagorean theorem holds.

Identify the components of the right-angled triangle.

Apply the Pythagorean theorem.

Visualize the setup and form a right-angled triangle.

Apply the Pythagorean theorem.

Identify the sides of the triangle and the longest side.

Check if the Pythagorean theorem holds.

Identify the components of the right-angled triangle formed by the diagonal.

Apply the Pythagorean theorem.

Identify the properties of an isosceles right triangle.

Apply the Pythagorean theorem.

Still stuck on this one?

More Mathematics Questions

Find the length of AD using the Pythagorean theorem in $\triangle ABD$ .

Find the length of $x$ (CD) using the Pythagorean theorem in $\triangle ADC$ .

Check if $3^2 + 7^2 = 9^2$ .

Check if $7^2 + 24^2 = 25^2$ .