Pythagoras Theorem — Multiple Choice Questions

In a △ABC, if AB = $6\sqrt{3}$ cm, BC = 6 cm and AC = 12 cm, then ∠B is

1. 120°

2. 90°

3. 60°

4. 45°

Answer

Here greatest length is 12 cm and other lengths are 6 cm, $6\sqrt{3}$ cm.

Note that 12² = 144 and 6² + $(6\sqrt{3})^2$ = 36 + 108 = 144.

Thus, 12² = 6² + $(6\sqrt{3})^2$.

Hence, ABC is right triangle with hypotenuse = AC = 12 cm.

So, angle opposite to AC i.e. ∠B = 90°.

Hence, Option 2 is the correct option.

If the sides of a rectangular plot are 15 m and 8 m, then the length of its diagonal is

1. 17 m

2. 23 m

3. 21 m

4. 17 cm

Answer

As sides of rectangle are perpendicular to each other so △ABC is a right angle triangle.

In right angle triangle ABC,

⇒ AC² = AB² + BC²

⇒ AC² = 8² + 15²

⇒ AC² = 64 + 225

⇒ AC² = 289

⇒ AC = $\sqrt{289}$ = 17 m.

Hence, Option 1 is the correct option.

The lengths of the diagonals of a rhombus are 16 cm and 12 cm. The length of the side of rhombus is

1. 9 cm

2. 10 cm

3. 8 cm

4. 20 cm

Answer

Let AC = 16 cm and BD = 12 cm.

We know that,

Diagonals of rhombus are perpendicular and bisect each other,

OB = $\dfrac{1}{2}BD$ = 6 cm and AO = $\dfrac{1}{2}$AC = 8 cm.

In right triangle AOB,

By pythagoras theorem we get,

⇒ AB² = AO² + OB²

⇒ AB² = 8² + 6²

⇒ AB² = 64 + 36

⇒ AB² = 100

⇒ AB = $\sqrt{100}$ = 10 cm.

Hence, Option 2 is the correct option.

If a side of a rhombus is 10 cm and one of the diagonals is 16 cm, then the length of the other diagonal is

1. 6 cm

2. 12 cm

3. 20 cm

4. 12 cm

Answer

Let AC = 16 cm.

We know that,

Diagonals of rhombus are perpendicular and bisect each other,

AO = $\dfrac{1}{2}$AC = 8 cm.

In right triangle AOB,

By pythagoras theorem we get,

⇒ AB² = AO² + OB²

⇒ 10² = 8² + OB²

⇒ 100 = 64 + OB²

⇒ OB² = 100 - 64 = 36

⇒ OB = $\sqrt{36}$ = 6 cm.

BD = 2OB = 12 cm.

Hence, Option 2 is the correct option.

If a ladder 10 m long reaches a window 8 m above the ground, then the distance of the foot of the ladder from the base of the wall is

1. 18 m

2. 8 m

3. 6 m

4. 4 m

Answer

Let AB be the ladder and B be the point of window.

In right triangle ACB,

By pythagoras theorem,

⇒ AB² = AC² + BC²

⇒ 10² = AC² + 8²

⇒ AC² = 100 - 64

⇒ AC² = 36

⇒ AC = $\sqrt{36}$ = 6 m.

Hence, Option 3 is the correct option.

A girl walks 200 m towards East and then she walks 150 m towards North. The distance of the girl from starting point is

1. 350 m

2. 250 m

3. 300 m

4. 225 m

Answer

Let A be starting point and B is the end point.

In right triangle ACB,

By pythagoras theorem we get,

AB² = AC² + CB²

AB² = (200)² + (150)²

AB² = 40000 + 22500

AB² = 62500

AB = $\sqrt{62500}$ = 250 m.

Hence, Option 2 is the correct option.

A ladder reaches a window 12 m above the ground on one side of the street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 9 m high. If the length of the ladder is 15 m, then the width of the street is

1. 30 m

2. 24 m

3. 21 m

4. 18 m

Answer

In right triangle AEB,

By pythagoras theorem we get,

⇒ EB² = EA² + AB²

⇒ 15² = 9² + AB²

⇒ 225 = 81 + AB²

⇒ AB² = 144

⇒ AB = $\sqrt{144}$ = 12 m

In right triangle BCD,

By pythagoras theorem we get,

⇒ BD² = BC² + CD²

⇒ 15² = BC² + 12²

⇒ 225 = BC² + 144

⇒ 225 - 144 = BC²

⇒ BC² = 81

⇒ BC = $\sqrt{81}$ = 9 m

⇒ AC = AB + BC = 12 + 9 = 21 m.

Hence, Option 3 is the correct option.

Consider the following two statements:

Statement 1: The area of a square whose diagonal is 6 cm is 36 cm².

Statement 2: A diagonal of a square divides it into two right angled isosceles triangle.

Which of the following is valid?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and Statement 2 is false.

4. Statement 1 is false, and Statement 2 is true.

Answer

Let's consider a square ABCD with diagonal AC.

A square is a quadrilateral with four right angles.

Therefore, ∠A = ∠B = ∠C = ∠D = 90°.

AC divides the square into two triangles: △ABC and △ADC.

Both these triangles contain a right angle (at B and D respectively). Thus, they are right-angled triangles.

A square has all four sides equal in length. So, AB = BC = CD = DA.

In △ABC, the two sides AB and BC are equal (sides of the square).

In △ADC, the two sides AD and CD are equal (sides of the square).

Thus, AC divides the square into two right angled isosceles triangles.

∴ Statement 2 is true.

In triangle ABC,

By the Pythagorean theorem:

⇒ AC² = AB² + BC²

Let length of each side of square be a cm and length of diagonal equal to 6 cm (given).

⇒ 6² = a² + a²

⇒ 36 = 2a²

⇒ a² = $\dfrac{36}{2}$ = 18 cm²

As we know that area of square = a²

Thus, area = 18 cm².

∴ Statement 1 is false.

∴ Statement 1 is false, and Statement 2 is true.

Hence, option 4 is the correct option.