Question

the semi perimeter of a right angle triangle is 24 cm if the hypotenuse is 20 cm what is the area of triangle​

Answer 1

Answer:

Area of the triangle is 96 cm².

Step-by-step explanation:

Let the height be 'x' and base be 'y'.  As the other side(hypotenuse) is 20 cm.

∴ Semi-perimeter = (x + y + 20)/2

          24 = (x + y + 20)/2    

          48 = x + y + 20

          28 = x + y         ...(1)

Given, hypotenuse = 20 cm.      Using Pythagoras theorem,

⇒ x² + y² = 20²

⇒ x² + y² = 400           ...(2)

       Square on both sides of (1):

⇒ (x + y)² = 28²

⇒ x² + y² + 2xy = 784

⇒ 400 + 2xy = 784      [from (2)]

⇒ 2xy = 384

⇒ xy = 192               ...(3)

       ∴ Area = 1/2 * height * base

                    = 1/2 * x * y

                    = 1/2 * xy

                    = 1/2 * 192       [from (3)]

                    = 96 cm²

Answer 2

Given :-

• The semi perimeter of a right angle triangle is 24 cm if the hypotenuse is 20 cm.

To Find :-

• What is the area of triangle?

Solution :-

• Let height of triangle be m
• And base of triangle be n

Here, it is given that hypotenuse of right angled triangle is 20 cm and we know that by pythagoras theorem;

• Hypotenuse² = Height² + Base²

Putting all values we get,

➡ 20² = m² + n²

➡ m² + n² = 400 • • • ①

Now, it is given that semi perimeter of triangle is 24 cm and we know that semi perimeter of triangle is given by;

• Semi perimeter = Perimeter/2

We can write it as;

• Semi perimeter = (Sum of all side of triangle)/2

Putting all values we get,

➡ (m + n + 20)/2 = 24

➡ m + n + 20 = 24 × 2

➡ m + n + 20 = 48

➡ m + n = 48 - 20

➡ m + n = 28

Squaring both side we get,

➡ (m + n)² = 28²

Using identity { (a + b)² = a² + b² + 2ab } in LHS we get,

➡ m² + n² + 2mn = 784 • • • ②

From ① put in ② we get,

➡ 400 + 2mn = 784

➡ 2mn = 784 - 400

➡ 2mn = 384

➡ mn = 384/2

➡ mn = 192

Now, we know that area of right angle triangle is given by;

• Area = ½ × Base × Height

Putting all values we get,

➡ Area = ½ × m × n

➡ Area = (m × n)/2

➡ Area = mn/2

Put value of mn we get,

➡ Area = 192/2

➡ Area = 96 cm²

• Hence, area of triangle is 96 cm².

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