Question

Find the lengths of the sides of a triangle if two of the sides are equal, the third side is 1 1/3 cm longer than the others, and its perimeter is 5 2/5 cm. The two equal sides of the triangle are (blank)cm. The third side is (blank) cm.

Answer 1

One equal side =
`1(16)/(45)`cm and third side is
`2(31)/(45)`cm

Explanation:

This is describing an isosceles triangle

1
`(1)/(3)` =
`(4)/(3)`

5
`(2)/(5)` =
`(27)/(5)`

Let

x = one of the two equal sides of the triangle

∴ Third side of triangle =
`(4)/(3) + x`

Perimeter of a triangle = Sum of all three sides:

`(27)/(5)`
`= x + x + ((4)/(3) + x)`

Expand the parenthesis using the Distributive Law and bring all the like terms together:

`=(27)/(5) = 2x + (4)/(3) + x`

`= (27)/(5) = 3x + (4)/(3)`

`= (27)/(5) -(4)/(3) = 3x`

The two denominators of the two fractions have to be manipulated to be made the same:

`= ((3)/(3))((27)/(5)) - ((5)/(5))((4)/(3)) = 3x`

`= (81)/(15) - (20)/(15) = 3x`

`= (81 - 20)/(15) = 3x`

`= (61)/(15) = 3x`

Cross-multiplication is added:

`= (61)(1) = (15)(3x)`

`= 61 = 45x`

Isolate x and make it the subject of the formula:

x =
`(61)/(45)`

x = One of the two equal sides =
`1(16)/(45)`cm

∴Third side:

=
`(61)/(45) + (4)/(3)`

=
`(61)/(45) + ((15)/(15))((4)/(3))`

=
`(61)/(45) + (60)/(45)`

=
`(61 + 60)/(45)`

=
`(121)/(45)`

=
`2(31)/(45)`cm

Answer 2

Explanation:

Here's a step by step solution with more details:

Let's call the length of the two equal sides of the triangle as x.

The third side, which is 1 1/3 cm longer, will have a length of x + 1 1/3 cm.

The perimeter of the triangle is 5 2/5 cm, so we can write an equation using the lengths of the sides:

x + x + (x + 1 1/3) = 5 2/5

Simplifying the equation:

2x + 1 1/3 = 5 2/5

Subtracting 1 1/3 from both sides:

2x = 4 1/5

Dividing both sides by 2:

x = 2 2/5

So the two equal sides of the triangle are 2 2/5 cm and the third side is 2 2/5 + 1 1/3 = 3 7/15 cm.