Question

⋆The perimeter of an isosceles triangle is 42cm and base is 1 ½ times the each of the equal sides.
⋆Find the length of each side of triangle, area of the triangle, the height of the triangle?

Answer 1

➢Question: The perimeter of an isosceles triangle is 42cm and base is 1 ½ times each of the equal sides:

➢To find:

⋆ Length of each side of the triangle

⋆ Area of the triangle

⋆ Height if the triangle

➢Answer:

➢Given:

✶ Perimeter of triangle = 42cm

✶ Base = 1½ times each of equal sides

➢ Solution:

٭ Let each side of equal sides be a cm and base be b cm

٭ Then we get, b = ³⁄₂a

٭ Perimeter = sum of the all sides of triangle

٭ Perimeter = a + a + b

٭ Perimeter = a + a + ³⁄₂a

✯ 42cm = a + a + ³⁄₂a

✯ 42 ×  ²⁄₇ = a

✯  a = 12cm

✯ Each equal side = 12cm

✯ Base =  ³⁄₂ × 12

✯ Base = 18cm

➛ Area of the isosceles triangle =  ¼ b √4a²-b²

➛ Area of the isosceles triangle  =  ¼ × 18 × √24²-18²

➛ Area of the isosceles triangle = ⁹⁄₂ × 15.87

➛ Area of the isosceles triangle = 71.42cm²

➡ Height of the isosceles triangle = √4a²-b²÷2

⇢ We know the value of √4a²-b², so substitute

➡ Height of the isosceles triangle = 15.87 ÷ 2

➡ Height of the isosceles triangle  = 7.94cm

✯ Length of each side of the triangle = 18cm

✯ Area of the triangle = 71.42cm²

✯ Height of the triangle = 7.94cm

Answer 2

Given:-

• Perimeter of the triangle = 42 cm
• Base = 1½ of each of the equal sides.

To Find:-

• The length of each sides
• Area of the triangle
• Height of the triangl

Assumption:-

• Let the two equal side be x
• third side = 1½ x = 3x/2

Solution:-

We know,

Perimeter of a triangle = Sum of all the three sides of the triangle.

We are given with the perimeter of the triangle,

Hence,

x + x + 3x/2 = 42

⇒ 2x + 3x/2 = 42

⇒ (4x + 3x)/2 = 42

⇒ 7x = 42 × 2

⇒ x = 84/7

⇒ x = 12

Therefore, sides of the triangle is as follows:-

• Equal sides = x = 12 cm
• Base = 3x/2 = (3 × 12)/2 = 3 × 6 = 18 cm

We can write:-

• 1st side (a) = 12 cm
• 2nd side (b) = 12 cm
• 3rd side (c) = 18 cm

Here, we got all the sides of the triangle.

So by applying Heron's formula we can find the area of the triangle.

Let us first find the semi - perimeter of the triangle.

We know,

• s = (a + b + c)/2

Hence,

s = (12 + 12 + 18)/2

⇒ s = 42/2

⇒ s = 21

According to Heron's formula,

• A = √s(s - a)(s - b)(s - c)

Hence,

Area = √21(21 - 12)(21 - 12)(21 - 18)

⇒ Area = √21 × 9 × 9 × 3

⇒ Area = √3 × 7 × 9 × 9 × 3

⇒ Area = 3 × 9√7

⇒ Area = 27√7

Taking √7 = 2.6

⇒ Area = 27 × 2.6

⇒ Area = 70.2

∴ Area of the triangle is 70.2 cm²

Here we got the area of the triangle.

We also have one formula for area of triangle we goes like this:-

• Area = 1/2 × base × height

Putting the area and base in this formula we'll get the height of the triangle.

Hence,

Area = 1/2 × base × height

⇒ 70.2 = 1/2 × 18 × h

⇒ 70.2 = 9h

⇒h = 70.2/9

⇒ h = 7.8

∴ Height of the triangle is 7.8 cm

________________________________