Circle Geometry: Arc Lengths

• Jul 4th 2010, 05:11 PM
Lukybear
Circle Geometry: Arc Lengths
If two chords of a circle AB and CD meet at right angles, show that the length of the arc AC plus the arc BD is equal to half the circumference of the circle.

Here i tried to prove that angle AOC and angle BOD = 180, where O is centre and hence arc lengths would be half the circumference. However i cannot do that.

I am pretty sure the approach of this question is using arc lengths and angles at centre.

However if any other approach is found that would be brilliant.
• Jul 4th 2010, 09:46 PM
Soroban
Hello, Lukybear!

Quote:

If two chords of a circle $AB$ and $CD$ meet at right angles,
show that arc AC + arc BD is equal to half the circumference of the circle.

Let the two chords intersect at $P.$
. . Then: . $\angle APC = 90^o.$

By the "intersecting chords theorem", we have:

. . $\angle APC \;=\;\dfrac{\text{arc}(AC) + \text{arc}(BD)}{2} \:=\:90^o$

Therefore: . $\text{arc}(AC) + \text{arc}(BD) \:=\:180^o$