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
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