Circles and Right Triangles

Circle Theorem

Theorem 1

The angle at the center of the circle is twice the angle at the circums standing on the same arc.

angle x = 2 multiply by angle y

or

angle y = angle x divided by 2

or

AOB = 2 ACB

Example

If AOB = 96 degrees . determine ACB.

x = 96 degrees

y = 96/2

= 48 degrees

Theorem 2

The angle in a semi circle is a right angle. The triangle passes though the circle of the circle and AC is a diameter of the circle. Hence the triangle is a right angle triangle.

ABC = 90 degrees

Example

Determine the length of the diameter AC.

using Pythagoras's Theorem:
  $AC^2=AB^2+BC^2$  
  $AC^2=(12)^2+(5)^2$  
  $AC^2=144+25$  
  $AC^2=169$  
AC = 13cm 

Theorem 3

Angles at the circumference of a circle standing on the same arc are equal.

ACB and ADB are angles at the circumference of the circle both standing on the same arc.

Therefore ACB = ADB

Example

If angle ACB = 59 degrees , determine the size of angle ADB.

ACB = 59 degrees

ACB = ADB = 59 degrees because the angles are on the same arc of the circumference.

Theorem 4

The opposite angles of a quadrilateral are supplementary. (angles that sum to 180 degrees)

angles A an C are opposite angles

Therefore A + C = 180 degrees

angles B and D are opposite

Therefore B + D = 180 degrees

Example

Without measuring, determine the values of the unknown angles in the circle.

opposite angles

x and P = 180 = 43 + x

Therefore x = 180 - 43 = 137 degrees

y and Q = 180 = 98 + y

Therefore x = 180 - 98 = 82 degrees

Theorem 5

The exterior angles of a quadrilateral is equal to the interior opposite angle.

WDC = B

XCB = A

YBA = D

ZAD = C

Example

Determine the magnitude of each of the marked angles in the circle.

s + 85 = 180 degrees (straight line)

s = 180 - 85

= 95 degrees

opposite interior angles sum to 180 degrees

Therefore Q + s =180

Q = 180 - 95

= 85 degrees