Question

jang
3. The perimeter of a right angle triangle is 12cm and it's
hypotenuse is 5cm Find both
sides and area of the triangle , verify the
result obtained by heroine formula ? ​

Answer 1

Given perimeter of the right angle triangle = 12cm

hypotenuse of the triangle = 5cm

let the all three sides of the right angle triangle be a, b and c respectively. where c is the hypotenuse of the right angle triangle.

we know that,

perimeter of a triangle = sum of all three sides

⇒ a + b + c = 12cm

⇒ a + b + 5 = 12cm

⇒ a + b = 12 - 5

⇒ a = 7 - b ------(i)

by pythegoras theorem, we get

⇒ c² = a² + b² ------(i)

substituting value of a from equation (i) in equation (ii)

⇒ 5² =  (7 - b)² + b²

⇒ 25 = (7)² - 2(7)(b) + (b)² + b²

⇒ 25 = 49 - 14b + 2b²

⇒ 2b² - 14b + 24 = 0

taking 2 as common,

⇒ b² - 7b + 12 = 0

by factorization, we get

⇒ b² - (4b + 3b) + 12 = 0

⇒ b² - 4b - 3b + 12 = 0

⇒ b(b - 4) - 3(b - 4) = 0

⇒ (b - 4) (b - 3)

∴ b = 4 or 3

if b = 4, then a = 7 - 4 = 3

if b = 3, then a = 7 - 3 = 4

so, we can say the the other two sides of the right angle triangle (base and perpendicular) are 3cm and 4cm respectively.

∴ area of the right angle triangle = 1/2 * b * h

= 1/2 * 3 * 4

= 3 * 2

= 6cm²

VERIFICATION BY HERON'S FORMULA :-

semi-perimeter = 12/2 = 6cm

$\sf area \: of\: the\: triangle = \sqrt{s(s-a)(s-b)(s-c)} \\ \\ \sf = \sqrt{6(6-3)(6-4)(6-5)} \\ \\ \sf = \sqrt{6*3*2*1} \\ \\ \sf=\sqrt{36} \\ \\ \sf = \boxed{6cm^{2} }$

hence verified!

Answer 2

Solution :-

Suppose x and y are length of other two sides of right triangle.

_______________________[Assume]

Perimeter of right angle triangle = 12 cm

Hypotenuse = 5 cm

_______________________[Given]

We know that,

perimeter of triangle = sum of all sides

⇒x + y + 5 = 12

⇒x + y = 12 - 5

⇒y = 7−x ......(i)

Using Pythagoras theorem we have ;

x² + y² = 5² ......(ii)

Putting the value of x in eq(ii), we get

x² + (7 − x)² = 5²

⇒x² + 49 + x² −14x = 25

⇒2x² − 14x + 24 = 0

⇒x² − 7x + 12 = 0

⇒x² − 4x −3x + 12 = 0

⇒x(x − 4) − 3(x − 4) = 0

⇒(x − 3)(x − 4) = 0

⇒x = 3 Or x = 4

When x = 3 cm , then y = 7 - 3 = 4 cm

When x = 4 cm , then y = 7 - 4 = 3 cm

say, base = b = 3 cm & height = h = 4 cm

Now,

Area of right angle triangle = 1/2 × b × h

= 1/2 × 3 × 4

= 3 × 2

= 6 cm²

______________________________

✡ Verification by Heron's Formula :-

Semi-perimeter of triangle = s = 12/2 = 6 cm

So, using heron's formula we have ;

Area of triangle = √[s(s − a)(s − b)(s − c)]

= √[6(6 − 3)(6 − 4)(6 − 5)]

= √(6 × 3 × 2 × 1)

= √36

= 6 cm²

★ Hence Verified —

_______________________________