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Question

The sides of certain triangles are given below. Determine which of them are right triangles.

(i) 9 cm, 16 cm, 18 cm

(ii) 7 cm, 24 cm, 25 cm

(iii) 1.4 cm, 4.8 cm, 5 cm

(iv) 1.6 cm, 3.8 cm, 4 cm

(v) (a-1) cm, 2acm,(a+1)cm

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Solution

For a given triangle to be a right-angled, the sum of the squares of the two sides must be equal to the square of the largest side.

(i) Let a = 9 cm, b = 16 cm and c = 18 cm. Then

(a2+b2)=[92+(16)2]=(81+256) cm2=337 cm2And c2=(18)2cm2=324 cm2

Therefore,
(a2+b2)c2

Hence, the given triangle is not right-angled.

(ii) Let a = 7 cm, b = 24 cm and c = 25 cm, Then

(a2+b2)=[72+(24)2]=(49+576) cm2=625 cm2And c2=(25)2cm2=625cm2

Therefore,
(a2+b2)=c2

Hence, the given triangle is a right triangle.

(iii) Let a = 1.4 cm, b = 4.8 cm, and c = 5 cm

(a2+b2)=[(1.4)2+(4.8)2]=(1.96+23.04) cm2=25 cm2And c2=(5)2cm2=25cm2

Therefore,
(a2+b2)=c2

Hence, the given triangle is a right triangle.

(iv) Let a = 1.6 cm, b = 3.8 cm and c = 4 cm

(a2+b2)=[(1.6)2+(3.8)2]=(2.56+14.44)cm2=17 cm2And c2=(4)2cm2=16 cm2

Therefore,
(a2+b2)c2

Hence, the given triangle is not a right triangle.

(v) Let p = (a - 1) cm, q = 2a cm and r = (a + 1) cm

(p2+q2)=[(a1)2+(2a)2]=(a2+12a+4a) cm2=(a+1)2 cm2And r2=(a+1)2 cm2

Therefore,
(p2+q2)=r2

Hence, the given triangle is a right triangle.


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