Question

In an isosceles triangle, the difference between one of the equal sides and the unequal side (longest of the three) is 3/10 of the sum of the equal sides. If the perimeter of the triangle is 90 cm, then find the length of unequal side in centimetres.​

Answer 1

Given information,

In an isosceles triangle, the difference between one of the equal sides and the unequal side (longest of the three) is 3/10 of the sum of the equal sides. If the perimeter of the triangle is 90 cm, then find the length of unequal side in centimetres.

• One equal side - Unequal side = 3/10(Sum of equal sides)
• Perimeter of triangle = 90 cm
• Length of unequal side = ?

Let,

• Equal sides of triangle = x cm

Now,

➻ Sum of equal sides = (x + x) cm

➻ Sum of equal sides = 2x cm

We know that,

• One equal side - Unequal side = 3/10(Sum of equal sides)

Difference of equal side and unequal side,

➻ Difference = 3/10(2x)

➻ Difference = 3/10 × 2 × x

➻ Difference = 3/5 × x

➻ Difference = 3x/5

• Hence, Difference between one of the equal sides and the unequal side is 3x/5 cm.

Let,

• Unequal side of triangle = y cm

According to the Question,

➻ x - y = 3x/5

➻ 5(x - y) = 3x

➻ 5x - 5y = 3x

➻ 5x - 5y - 3x = 0

➻ 5x - 3x - 5y = 0

➻ 2x - 5y = 0 ⠀⠀⠀⠀⠀⠀⠀⠀– (1)

Also,

➻ Perimeter of triangle = 90 cm

• Perimeter of ∆ = Sum of it's all sides

➻ x + x + y = 90

➻ 2x + y = 90

➻ 2x + y - 90 = 0 -(2)

Subtracting (2) from (1),

➻ 2x - 5y - (2x + y - 90) = 0

➻ 2x - 5y - 2x - y + 90 = 0

➻ 2x - 2x - 5y - y + 90 = 0

➻ 0 - 6y + 90 = 0

➻ - 6y + 90 = 0

➻ - 6y = 0 - 90

➻ - 6y = - 90

➻ 6y = 90

➻ y = 90/6

➻ y = 15

• Hence, length of unequal side of triangle is 15 cm.

Finding value of x,

➻ 2x - 5y = 0⠀⠀⠀⠀⠀⠀⠀⠀ [From (1)]

Putting value of y,

➻ 2x - 5(15) = 0

➻ 2x - (5 × 15) = 0

➻ 2x - 75 = 0

➻ 2x = 0 + 75

➻ 2x = 75

➻ x = 75/2

➻ x = 37.5

Verification,

➻ Perimeter of triangle = 90

➻ x + x + y = 90

➻ 37.5 + 37.5 + 15 = 90

➻ 75 + 15 = 90

➻ 90 = 90

➻ LHS = RHS

• Hence, Verified ✔

Know more,

• Perimeter of any figure is calculated by sum of it's all sides.
• Perimeter of square = 4 × side
• Perimeter of equilateral ∆ = 3 × side
• Perimeter of circle = 2πr
• Perimeter of rectangle = 2(L + B)
• Perimeter of rhombus = 4 × side

▬▬▬▬▬▬▬▬▬▬▬▬

Answer 2

Let :-

• The equal side of ∆ = x
• The unequal side of ∆ = y

To Find :-

• The length of unequal side of ∆ = ?

Solution :-

• To calculate the unequal side of isoceles ∆ at first we have to set up equation then calculate it's side by solving the equation :-

Calculation for 1st equation :-

Difference (unequal side - equal side) = 3/10 of (sum of equal side)

⇢ (y - x) = 3/10 × (x + x)

⇢ y - x = 3/10 × 2x

⇢ y - x = 3x/5

⇢ 5y - 5x = 3x

⇢ 5x + 3x - 5y = 0

⇢ 8x - 5y = 0--------(i)

Calculation for 2nd equation :-

The perimeter of the triangle = 90cm

⇢ Sum of equal and unequal side = 90cm

⇢ (x + x) + y = 90

⇢ 2x + y = 90--------(ii)

Now in equation (i) × 2 (ii) × 8 subtracting we get :-]

⇢ 16x - 10y = 0

⇢ 16x + 8y = 720

By solving we get here :-]

⇢ -18y = - 720 ⇢y = 40 cm

Putting the value of y = 15 in eq (I) we get

⇢ 8x - 5y = 0

⇢ 8x - 5(40) = 0

⇢ 8x = 200 ⇢x = 25 cm

Hence, The length of unequal side = 40 cm

Verification :-

⇢ Perimeter of the triangle = 90

⇢ (x + x) + y = 90

⇢ (25 + 25) + 40 = 90

⇢ 50 + 40 = 90

⇢ 90 = 90

Hence , verified :-