Square outiside circle outside quadrilateral

This was a problem on my brothers maths test, and it is doing my families collective heads in.

A quadrilateral with three sides measuring 15cm and one side 20cm sits within a circle with each of its four corners touching the circle.

Similarly, a square sits around the outside of the circle, touching in the middle of each side. What is the area of the square?

Also, the problem had to be solved without a calculator.

1 Attachment(s)

Re: Square outiside circle outside quadrilateral

see attached diagram ...

1. trapezoid diagonal bisects the base angle

2. notice small right triangle formed by trapezoid height ...

3. inscribed angle is half the central angle

4. using law of cosines ...

5. area of large square (not drawn) is ...

Re: Square outiside circle outside quadrilateral

I constructed the regular trapezoid and found the radius of the circumcircle to be 11.5 cm. side of square 23cm. area of square 529 cm^2. No calculator.

Re: Square outiside circle outside quadrilateral

Quote:

Originally Posted by

**bjhopper** I constructed the regular trapezoid and found the radius of the circumcircle to be 11.5 cm. side of square 23cm. area of square 529 cm^2. No calculator.

a 2% error ... not bad.

;)

Re: Square outiside circle outside quadrilateral

thanks Skeeter, test was without a calculator, did you manage that?

Re: Square outiside circle outside quadrilateral

thanks BJ, we did that too on the computer in publisher (? i know) and got a diameter of 23.306, area of square 543.16963. Still, test was without rulers or calculator. Test was actually an Australia wide maths comp for year 10 students.

Re: Square outiside circle outside quadrilateral

Quote:

Originally Posted by

**Linus** thanks Skeeter, test was without a calculator, did you manage that?

did you read my solution? no calculator required.