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The difference of length s of sides forming right angled triangle is 3 cm. If the perimeter of the triangle is 36 cm. Find the area of the triangle

Answers

Answered by Dalfon
210

Question:

The difference of lengths of sides forming right angled triangle is 3 cm. If the perimeter of the triangle is 36 cm. Find the area of the triangle

Answer:

Area of triangle = 54 cm²

Step-by-step explanation:

Given that the difference of lengths of sides forming right angled triangle is 3 cm and the perimeter of the triangle is 36 cm. We need to find out the area of the triangle.

Let's say that the sides of the triangle are "x, y and z."

Now, the perimeter of the triangle is 36 cm or sum of all sides of the triangle is 36 cm.

→ x + y + z = 36

→ x + y = 36 - z

Square both the sides,

→ (x + y)² = (36 - z)²

→ x² + y² + 2xy = 1296 + z² - 72z ----------(eq 1)

Used identity: (a + b)² = a² + b² + 2ab

Also said that the difference of the lengths of sides forming right angles triangle is 3 cm. Let's say the bigger side i.e. hypotenuse be "z" and perpendicular and base be "x and y." So,

→ x - y = 3 ----------(eq A)

Square both the sides

→ (x - y)² = (3)²

→ x² + y² - 2xy = 9 -----------(eq 2)

Used identity: (a - b)² = a² + b² - 2ab

On adding (eq 1) & (eq 2) we get,

→ 2x² + 2y² = 1305 + z² - 72z

From Pythagoras theorem we can say that: x² + y² = z²

→ 2(x² + y²) = 1305 + z² - 72z

→ 2z² - z² = 1305 - 72z

→ z² + 72z - 1305 = 0

The above equation is in the form x² + bx + c = 0. So, use the quadratic formula to solve it. Where, a is 1, b is 72 and c is -1305.

Quadratic formula: z = [-b ±√(b² - 4ac)]/2a. On solving we get,

→ z = 15, -87 (neglect the negative one)

→ z = 15 (hypotenuse)

Therefore,

→ x + y + 15 = 36

→ x + y = 36 - 15

→ x + y = 21 -----------(eq 3)

On adding (eq A) and (eq 3) we get,

→ 2x = 24

→ x = 12 (perpendicular)

Substitute value of x in (eq A)

→ x - y = 3

→ 12 - 3 = y

→ 9 = y (base)

Method 2: As assumed above that perpendicular of the triangle is "x" cm. So, base of the triangle is "3 + x" cm.

→ Hypotenuse + Perpendicular + Base = Perimeter of triangle

→ Hypotenuse + x + 3 + x = 33

→ Hypotenuse + 2x = 33

→ Hypotenuse = 33 - 2x

From above we have the value of hypotenuse, perpendicular and base. On applying Pythagoras theorem we get,

→ x² - 69x + 540 = 0

On solving we get,

→ x = 60 cm, 9 cm

Perimeter of the triangle is 36 cm & 60 > 36. So, we can't take 60 as perpendicular. Base = 3 + 9 = 12 cm

Hence,

Area of triangle = 1/2 × base × perpendicular (or height)

= 1/2 × 9 × 12

= 9 × 6

= 54 cm²

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