Square outiside circle outside quadrilateral

**Linus** · September 19th 2012, 03:42 PM

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.

**skeeter** · September 19th 2012, 05:14 PM

see attached diagram ...

1. trapezoid diagonal bisects the base angle

2. notice small right triangle formed by trapezoid height ... $\cos(2x) = \frac{2.5}{15} = \frac{1}{6}$

3. inscribed angle is half the central angle

4. using law of cosines ...

$15^2 = r^2 + r^2 - 2r^2\cos(2x) = 2r^2[1 - \cos(2x)]$

$2r^2 = \frac{15^2}{1 - \cos(2x)}$

$2r^2 = \frac{15^2}{1 - \frac{1}{6}}$

$2r^2 = \frac{6 \cdot 15^2}{5}$

5. area of large square (not drawn) is $4r^2$ ...

$4r^2 = \frac{2 \cdot 6 \cdot 15^2}{5} = 540$

**bjhopper** · September 19th 2012, 06:49 PM

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.

**skeeter** · September 20th 2012, 03:02 PM

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.

**Linus** · September 20th 2012, 04:25 PM

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

**Linus** · September 20th 2012, 04:33 PM

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.

**skeeter** · September 20th 2012, 05:07 PM

Originally Posted by Linus

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

did you read my solution? no calculator required.

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