circle geometry theorem 1

Theorem The line that connects the midpoint of a circle with the midpoint of a chord is perpendicular to the chord. Given: Circle with midpoint O and chord XY XM = MY To prove: OM  XY Proof:

Construct OX and OY

In OXM and OYM: i) OX = OY ii) OM = OM iii) XM = MY  OXM  OYM  M1 = M2 but M1 + M2 = 180°  M1 = M2 = 90°  OM  XY

radiuses common side given SSS follows from congruency angles on a straight line

Inverse The perpendicular line from the midpoint of a circle to any chord bisects the chord Given: Circle with midpoint O and OM  XY To prove:

XM = MY

Proof:

Construct OX en OY

Geometry: Circles

In OXM and OYM:



i) OX = OY radiuses ii) OM = OM common side iii M1 = M2 = 90° given OXM  OYM 90°Hyp S



XM = MY

follows from congruency

After learning about the context and meaning of these two theorems we could then make the following deductions: The perpendicular bisector of a chord will pass through the midpoint of a circle. If we draw three chords, and we construct the perpendicular bisector of each, these lines will intersect each other at exactly the midpoint of the circle. If, therefore, we have any three points that do not lie in a straight line, we should be able to draw a circle that will go through each of these points, and such a circle will be known as the circumscribed circle

Geometry: Circles

Examples Give reasons with each of the following problems. Round your answers to two decimals where applicable. 1)

2)

In circle M PQ = 10, MT = 12 and TM  PQ. Calculate a) PT b)

3)

O is the midpoint of the circle. Calculate the length of AB.

The length of the radius

In circle M KT = 8, KL = LT and MK = 5. How long is LQ?

Geometry: Circles