# Thread: Circle Geometry: Arc Lengths

1. ## 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.

2. Hello, Lukybear!

If two chords of a circle $\displaystyle AB$ and $\displaystyle CD$ 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 $\displaystyle P.$
. . Then: .$\displaystyle \angle APC = 90^o.$

By the "intersecting chords theorem", we have:

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

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