Q. 184.0( 4 Votes )

# Length of hypotenuse and perimeter of a right-angle triangle are 13 cm and 30 cm. Let us write by calculating the area of triangle.

Answer :

Let the base, perpendicular and Hypotenuse of right-angled triangle be b, p and h respectively.

We know, Perimeter of right-angled triangle = b + p + h

Given, perimeter = 30 cm

⇒ b + p + h = 30

⇒ b + p + 13 = 30 [∵ h, hypotenuse = 13 cm]

⇒ b = 17 – p [1]

Also, we know By Pythagoras theorem in right-angled triangle

(Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2}

⇒ h^{2} = p^{2} + b^{2}

⇒ (13)^{2} = p^{2} + (17 – p)^{2}

⇒ 169 = p^{2} + p^{2} + 289 – 34p

⇒ 2p^{2} – 34p + 120 = 0

⇒ p^{2} – 17p + 60 = 0

⇒ p^{2} – 12p - 5p + 60 = 0

⇒ (p – 12)(p - 5) = 0

⇒ p = 12 cm or p = 5 cm

Case I: p = 12 cm

⇒ b = 17 – 12 = 5 cm

and we know, area of triangle

Case II: p = 5 cm

⇒ b = 17 – 5 = 12 cm

and we know, area of triangle

Therefore, Area of triangle is 30 cm^{2}.

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