Question

An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of `18` and the triangle has an area of `64`. What are the lengths of sides A and B?

Answer 1

`11.47`

Explanation:

We can use the Heron's formula for area of a Triangle

`color(blue)("Area"=sqrt(s(s-a)(s-b)(s-c))`

Where,

`color(orange)("a, b and c are the sides"`

`color(orange)("s = semi-perimeter"=(a+b+c)/2`

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Let's assign the values

`color(purple)(a=x`

`color(purple)(b=x`

`color(purple)(c=18`

`color(purple)(s=(x+x+18)/2=(2x+18)/2`

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Let's start to solve

The whole equation starts here

`rArrcolor(violet)(sqrt((2x+18)/2((2x+18)/2-x)((2x+18)/2-x)((2x-18)/2-18))=64`

The whole equation equals `64` as, the area is `64`

`rarrsqrt((2x+18)/2((2x+18-2x)/2)((2x+18-2x)/2)((2x+18-36)/2))=64`

`rarrsqrt((2x+18)/2((color(red)(2x)+18-color(red)(2x))/2)((color(red)(2x)+18-color(red)(2x))/2)((2x-18-36)/2))=64`

`rarrsqrt((2x+18)/2((color(red)(2x-2x)+18)/2)((color(red)(2x-2x)+18)/2)((2x-18-36)/2))=64`

`rarrsqrt((2x+18)/2((color(red)(18))/2)((color(red)(18))/2)((2x-18-36)/2))=64`

`rarrsqrt((2x+18)/2(color(red)(9))(color(red)(9))((2x+18-36)/2))=64`

`rarrsqrt((2x+18)/2(color(red)(81))((2x+18-36)/2))=64`

`rarrsqrt((2x+18)/2(81)((2x+color(pink)(18-36))/2))=64`

`rarrsqrt((2x+18)/2(81)((2xcolor(pink)(-18))/2))=64`

Square both sides to take out the radical sign

`rarr(sqrt((2x+18)/2(81)((2x-18)/2)))^2=64^2`

`rarr((2x+18)/2)(81)((2x-18)/2)=color(red)(64*64)`

`rarr((2x+18)/2)(81)((2x-18)/2)=color(red)(4096)`

`rarrcolor(indigo)(((2x+18)/2)((2x-18)/2))(81)=color(red)(4096)`

Take out the common factor `2`

`rarrcolor(indigo)(((2(x+9))/2)((2(x-9))/2))(81)=color(red)(4096)`

`rarrcolor(indigo)(((cancel2(x+9))/cancel2)((cancel2(x-9))/cancel2))(81)=4096`

`rarrcolor(indigo)((x+9)(x-9))(81)=4096`

Use the indentity `color(brown)((a+b)(a-b)=a^2-b^2`

`rarrcolor(indigo)((x^2-9^2))(81)=4096`

`rarr(x^2-81)(81)=4096`

`rarr81x^2-81*81=4096`

`rarr81x^2-6561=4096`

`rarr81x^2=4096+6561`

`rarr81x^2=10657`

`rarrx^2=10567/81`

`rarrx^2=131.56`

Take the square root of both sides

`rarrsqrt(x^2)=sqrt(131.56)`

`color(green)(rArrx=11.47`