1
Question
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
(i) 8, 15, 17
(ii) 11, 12, 15
(iii) 11, 60, 61
(iv) 1.5, 1.6, 1.7
(v) 40, 20, 30
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Solution
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.
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.
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