# How to describe a circle about a rectangular triangle

Triangle - the simplest of flat polygonal figures. If the size of any corner is equal in its tops 90 °, then the triangle is called rectangular. About such polygon it is possible to draw a circle in such a way that each of three tops had one general point with its border (circle). This circle will be called described, and existence of a right angle considerably simplifies a problem of its construction.

## It is required to you

• Ruler, compasses, calculator.

## Instruction

1. Begin with definition radiusof circles which should be constructed. If there is an opportunity to measure lengths of the parties of a triangle, then pay attention to its hypotenuse - the party lying opposite to a right angle. Measure it and halve the received value - it is and there will be a radius of the circle described about a rectangular triangle.

2. If length of a hypotenuse is unknown, but there are lengths (an and b) legs (two parties adjacent to a right angle), radius (R) find with use of Pythagorean theorem. From it follows that this parameter will be equal to a half of the square root taken from the sum of the squared lengths of legs: R=½ * √ (a²+b²).

3. If length of only one of legs (a) and size of the acute angle adjoining to it is known (β), then for determination of radius of a circumscribed circle (R) use trigonometrical function - a cosine. In a rectangular triangle it defines a ratio of lengths of a hypotenuse and this leg. Calculate a half of a leg, private from division of length, into a cosine of the known corner: R=½*a/cos(β).

4. If except length of one of legs (a) the size of the acute angle (α) lying opposite to it is known, then for calculation of radius (R) use other trigonometrical function - a sine. Except replacement of function and the party in a formula nothing will change - divide leg length into a sine of the known acute angle, and divide result in half: R=½*b/sin(α).

5. After finding of radius determine by any of the listed ways the center of the described circle. For this purpose postpone the received value on compasses and establish it in any top of a triangle. To describe a cycle there is no need, just note the place of its suppression with a hypotenuse - this point and will be the center of a circle. Such is property of a rectangular triangle - the center of the circle described about it always is in the middle of its longest party. Draw a circle of the radius postponed on compasses with the center in the found point. On it construction will be complete.

Author: «MirrorInfo» Dream Team