# Mathematics Questions and Answers – Pythagoras Theorem

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Pythagoras Theorem”.

1. ∆ABC is a right angled triangle, where AB = 5cm, BC = 10cm, AC = 15cm.
a) False
b) True

Explanation: If ∆ABC is a right angled triangle, then it should satisfy the Pythagoras Theorem.
AB = 5cm, BC = 10cm, AC = 15cm
AB2 + BC2 = 52 + 102 = 25 + 100 = 125
AC2 = 152 = 225
Since, AC2 ≠ AB2 + BC2
Hence, ABC is not a right-angled triangle.

2. A man travels A to B, B to C, C to D and then finally D to E. What will be the shortest route the man could have taken?

a) 9.3 m
b) 9.2 m
c) 9.1 m
d) 9 m

Explanation: Here, there are two right-angled triangles: ∆ABC, ∆CDE
In ∆ABC, right-angled at B
AB2 + BC2 = 42 + 32 = 16 + 9 = 25 = AC2
AC = √25 = 5 m
CD2 + DE2 = 0.92 + 42 = 0.81 + 16 = 16.81 = EC2
EC = √16.81 = 4.1 m
The shortest distance the man should have travelled is 5 + 4.1 = 9.1 m

3. What will be the distance of the foot of ladder from the building, if the ladder of 12 m high reaches the top of a building 35 m high from the ground?
a) 32.65 m
b) 32.87 m
c) 31.87 m
d) 32.85 m

Explanation:

Here, AB is the building and AC is the ladder.
Now, the BC can be found out by Pythagoras Theorem.
AC2 = AB2 + BC2
352 = 122 + BC2
1225 = 144 + BC2
BC2 = 1225 – 144 = 1081
BC = √1081 = 32.87 m
The distance between the foot of the ladder and the building is 32.87 m.

4. The heights of two vertical lamp posts are 33 m and 24 m high. If the distance between them is 40 m, then what will be the distance between their tops?
a) 47.89m
b) 56.56m
c) 32.81m
d) 41m

Explanation:

Here, DC and AB are two lampposts of height 24 m and 33 m respectively.
The distance BC is 40m.
Now, draw a line perpendicular to AB from D.

Now, AED is a right-angled triangle, right angled at E.
AB = AE + EB
33 = AE + DC
33 = AE + 24
33 – 24 = AE
AE = 9m
In ∆AED,
AD = √1681 = 41 m
The distance between the two lampposts is 41 m.

5. ∆ABC is a right-angled triangle, right angled at B and BD ⊥ AC. If BD = 10cm, AB = 5 cm and BC = 5 cm then AC will be?
a) 44.72 cm
b) 5.59 cm
c) 18.11 cm
d) 22.36 cm

Explanation: The figure according to the given data is:

BD = 10cm and AB = 5cm
AD = √125 = 11.18 cm
Now, in ∆ABD and ∆CBD
BD = BD     (Common Side)
AB = BC     (Given)
∆ABD ≅ ∆CDB (RHS Congruency)
AC = AD + DC = 2AD = 2 × 11.18cm = 22.36cm

6. Which of triangle whose sides are given below are right angled?
a) AB = 89, AC = 80, BC = 39
b) AB = 57, AC = 50, BC = 45
c) AC = 34, AB = 20, BC = 21
d) AC = 50, AB = 32, BC = 20

Explanation: In (a), applying Pythagoras Theorem,
AC2 + BC2 = 802 + 392 = 7921 = AB2
Hence, this is a right angled triangle.
Now, in (b) applying Pythagoras Theorem,
AC2 + BC2 = 502 + 452 = 4525 ≠ AB2
Hence, this is not a right angled triangle.
Now, in (c) applying Pythagoras Theorem,
AB2 + BC2 = 202 + 212 = 841 ≠ AC2
Hence, this is not a right angled triangle.
Now, in (d) applying Pythagoras Theorem,
AB2 + BC2 = 322 + 202 = 1424 ≠ AC2
Hence, this is not right angled triangle.

7. The lengths of diagonals of a rhombus are 10 cm and 8 cm. What will be the length of the sides of rhombus?
a) 6.40 cm
b) 5.25 cm
c) 2.44 cm
d) 3.29 cm

Explanation:

ABCD is a rhombus. The length of diagonal is 10 cm and the length of other diagonal is 8 cm.
Since, diagonals of a rhombus bisect each other. Therefore, AE = 4cm and DE = 5 cm
Now, in ∆AED
AD2 = 42 + 5 2     (Since, AD is the altitude of the triangle it will bisect BC)
AD = √41 cm = 6.40 cm

8. If the side of rhombus is 13 cm and one of its diagonals is 24 cm, then what will be length of the other diagonal?
a) 8.4 cm
b) 4 cm
c) 11 cm
d) 10 cm

Explanation:

ABCD is a rhombus. The side of the rhombus is 13 cm and the length of one of its diagonal is 24 cm.
Let the length of other diagonal be 2x cm.
Since, diagonals of a rhombus bisect each other. Therefore, AE = x cm and DE = 12 cm
Now, in ∆AED
132 = x2 + 122     (Since, AD is the altitude of the triangle it will bisect BC)
x2 = 169 – 144
x2 = 25
x = √25 = 5 cm
AC = 2 × AE = 2 × 5 = 10 cm

9. What will be the length of the altitude of an equilateral triangle whose side is 9 cm?
a) 4.567 cm
b) 7.794 cm
c) 8.765 cm
d) 4.567 cm

Explanation:

Here, ABC is an equilateral triangle and AD is the altitude of the triangle.
92 = AD2 + 4.52     (Since, AD is the altitude of the triangle it will bisect BC)
AD = √60.75 = $$\frac {9\sqrt {3}}{2}$$ cm = 7.794 cm

10. What will be the length of the square inscribed in a circle of radius 5 cm?
a) 2.34 cm
b) 3.45 cm
c) 5√2 cm
d) 2.45 cm

Explanation:
The diagram according to the given data is:

ABCD is a square inscribed in a circle of radius 5 cm.
Now, joining the diagonals of the square we get

The diagonals intersect at E. We know that the diagonals of square are perpendicular to each other.
In ∆AED, using Pythagoras Theorem,
DE and EA are the radius of the circle, ∴ DE = EA
AD2 = 2 × 52 = 2 × 25 = 50