Ch.13.Pythagoras' Theoram

Question 1:

In the figures below, find the value of 'x'.

ANSWER:

(i)
  
In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(LM)² + l(MN)²
⇒(x)² = (7)² + (24)²
⇒x² = 49 + 576
⇒x² = 625
⇒x² = (25)²
⇒x = 25
∴ the value of x is 25.

(ii)

In the right angled triangle PQR, ∠Q = 90^∘. Hence, side PR is the hypotenuse.
According to Pythagoras' theorem,
l(PR)² = l(QR)² + l(PQ)²
⇒(41)² = (x)² + (9)²
⇒1681 = x² + 81
⇒x² = 1681 − 81
⇒x² = 1600
⇒x² = (40)²
⇒x = 40
∴ the value of x is 40.

(iii)

In the right angled triangle EDF, ∠D = 90^∘. Hence, side EF is the hypotenuse.
According to Pythagoras' theorem,
l(EF)² = l(ED)² + l(DF)²
⇒(17)² = (x)² + (8)²
⇒289 = x² + 64
⇒x² = 289 − 64
⇒x² = 225
⇒x² = (15)²
⇒x = 15
∴ the value of x is 15.

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Question 2:

In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.

ANSWER:

In the right angled triangle PQR, ∠P = 90^∘. Hence, side QR is the hypotenuse.
According to Pythagoras' theorem,
l(QR)² = l(PQ)² + l(PR)²
⇒l(QR)² = (24)² + (10)²
⇒l(QR)² = 576 + 100
⇒l(QR)² = 676
⇒l(QR)² = (26)²
⇒l(QR) = 26
∴ Length of seg QR = 26 cm.

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Question 3:

In the right-angled ∆LMN, ∠ M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.

ANSWER:

In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(MN)² + l(LM)²
⇒(20)² = l(MN)² + (12)²
⇒400 = l(MN)² + 144
⇒l(MN)² = 400 − 144
⇒l(MN)² = 256
⇒l(MN)² = (16)²
⇒l(MN) = 16
∴ Length of seg MN = 16 cm.

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Question 4:

The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder ?

ANSWER:

Let LN be ladder of length 15 m that is resting against a wall. Let M be the base of the wall and L be the position of the window.
The window is 9 m above the ground. Now, MN is the distance between base of the wall and that of the ladder.
In the right angled triangle LMN, ∠M = 90^∘. Hence, side LN is the hypotenuse.
According to Pythagoras' theorem,
l(LN)² = l(MN)² + l(LM)²
⇒(15)² = l(MN)² + (9)²
⇒225 = l(MN)² + 81
⇒l(MN)² = 225 − 81
⇒l(MN)² = 144
⇒l(MN)² = (12)²
⇒l(MN) = 12
∴ Length of seg MN = 12 m.
Hence, the distance between base of the wall and that of the ladder is 12 m.

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Question 1:

Find the Pythagorean triplets from among the following sets of numbers.
 (i) 3, 4, 5
(ii) 2, 4, 5
(iii) 4, 5, 6
(iv) 2, 6, 7
(v) 9, 40, 41
(vi) 4, 7, 8

ANSWER:

It is known that, if in a triplet of natural numbers, the square of the biggest number is equal to sum of the squares of the other two numbers, then the three numbers form a Pythgorean triplet.

(i) The given set of numbers is (3, 4, 5).
The biggest number among the given set is 5.
5² = 25; 4² = 16; 3² = 9
Now, 16 + 9 = 25
∴ 4² + 3² = 5²
Thus, (3, 4, 5) forms a Pythagorean triplet.

(ii) The given set of numbers is (2, 4, 5).
The biggest number among the given set is 5.
5² = 25; 4² = 16; 2² = 4
Now, 16 + 4 = 20 ≠ 25
∴ 4² + 2² ≠  5²
Thus, (2, 4, 5) does not form a Pythagorean triplet.

(iii) The given set of numbers is (4, 5, 6).
The biggest number among the given set is 6.
6² = 36; 5² = 25; 4² = 16
Now, 25 + 16 = 41 ≠ 36
∴ 5² + 4² ≠  6²
Thus, (4, 5, 6) does not form a Pythagorean triplet.

(iv) The given set of numbers is (2, 6, 7).
The biggest number among the given set is 7.
7² = 49; 6² = 36; 2² = 4
Now, 4 + 36 = 40 ≠ 49
∴ 2² + 6² ≠  7²
Thus, (2, 6, 7) does not form a Pythagorean triplet.

(v) The given set of numbers is (9, 40, 41).
The biggest number among the given set is 41.
9² = 81; 40² = 1600; 41² = 1681
Now, 81 + 1600 = 1681
∴ 9² + 40² = 41²
Thus, (9, 40, 41) forms a Pythagorean triplet.

(vi) The given set of numbers is (4, 7, 8).
The biggest number among the given set is 8.
8² = 64; 7² = 49; 4² = 16
Now, 16 + 49 = 65 ≠ 64
∴ 4² + 7² ≠  8²

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Question 2:

The sides of some triangles are given below. Find out which ones are right-angled triangles?
(i) 8, 15, 17
(ii) 11, 12, 15
(iii) 11, 60, 61
(iv) 1.5, 1.6, 1.7
(v) 40, 20, 30

ANSWER:

It is known that, if in a triplet of natural numbers, the square of the biggest number is equal to sum of the squares of the other two numbers, then the three numbers form a Pythgorean triplet.

If the lengths of the sides of a triangle form such a triplet, then the triangle is right angled triangle.

(i) The sides of the given triangle is 8, 15 and 17.
Let us check whether the given set (8, 15, 17) forms a Pythagorean triplet or not.
The biggest number among the given set is 17.
(17)² = 289; 

(15)² = 225; (8)² = 64
Now, 225 + 64 = 289
∴ (15)² + (8)² = (17)²
Thus, (8, 15, 17) forms a Pythagorean triplet.
Hence, the given triangle with sides 8, 15 and 17 is a right-angled triangle.

(ii) The sides of the given triangle is 11, 12 and 15.
Let us check whether the given set (11, 12, 15) forms a Pythagorean triplet or not.
The biggest number among the given set is 15.
(15)² = 225; (11)² = 121; (12)² = 144
Now, 121 + 144 = 265 ≠ 225
∴ (11)² + (12)² ≠ (15)²
Thus, (11, 12, 15) does not form a Pythagorean triplet.
Hence, the given triangle with sides 8, 15 and 17 is not a right-angled triangle.

(iii) The sides of the given triangle is 11, 60 and 61.
Let us check whether the given set (11, 60, 61) forms a Pythagorean triplet or not.
The biggest number among the given set is 61.
(61)² = 3721; (11)² = 121; (60)² = 3600
Now, 121 + 3600 = 3721
∴ (11)² + (60)² = (61)²
Thus, (11, 60, 61) forms a Pythagorean triplet.
Hence, the given triangle with sides 11, 60 and 61 is a right-angled triangle.

(iv) The sides of the given triangle is 1.5, 1.6 and 1.7.
Let us check whether the given set (1.5, 1.6, 1.7) forms a Pythagorean triplet or not.
The biggest number among the given set is 1.7.
(1.7)² = 2.89; (1.5)² = 2.25; (1.6)² = 2.56
Now, 2.25 + 2.56  = 4.81 ≠ 2.89
∴ (1.5)² + (1.6)² ≠ (1.7)²
Thus, (1.5, 1.6, 1.7) does not form a Pythagorean triplet.
Hence, the given triangle with sides 1.5, 1.6 and 1.7 is not a right-angled triangle.

(v) The sides of the given triangle is 40, 20 and 30.
Let us check whether the given set (40, 20, 30) forms a Pythagorean triplet or not.
The biggest number among the given set is 40.
(40)² = 1600; (20)² = 400; (30)² = 900
Now, 400 + 900 = 1300 ≠ 1600
∴ (20)² + (30)² ≠ (40)²
Thus, (40, 20, 30) does not form a Pythagorean triplet.
Hence, the given triangle with sides 40, 20 and 30 is not a right-angled triangle.