Mathematics
The sides of a right-angled triangle containing the right angle are 5x cm and (3x – 1) cm. Calculate the length of the hypotenuse of the triangle if its area is 60 cm2.
Mensuration
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Answer
Consider △ABC as a right angled triangle.
AB = 5x cm and BC = (3x – 1) cm
We know that,
Area of △ABC = × base × height = × BC × AB
Substituting the values we get,
⇒ 60 = × (3x – 1) × 5x
⇒ 120 = 5x(3x – 1)
⇒ 120 = 15x2 – 5x
⇒ 15x2 – 5x – 120 = 0
⇒ 5(3x2 – x – 24) = 0
⇒ 3x2 – x – 24 = 0
⇒ 3x2 – 9x + 8x – 24 = 0
⇒ 3x(x – 3) + 8(x - 3) = 0
⇒ (3x + 8)(x - 3) = 0
⇒ 3x + 8 = 0 or x - 3 = 0
⇒ 3x = -8 or x = 3
⇒ x = or x = 3
Since, x cannot be negative as length of a side cannot be negative. So, x = 3.
AB = 5 × 3 = 15 cm
BC = (3 × 3 – 1) = 9 – 1 = 8 cm
In right angled △ABC,
Using Pythagoras theorem,
⇒ AC2 = AB2 + BC2
Substituting the values we get,
⇒ AC2 = 152 + 82
⇒ AC2 = 225 + 64 = 289
⇒ AC2 = 172
So, AC = 17 cm.
Hence, the hypotenuse of the right angled triangle is 17 cm.
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