QC&DI book, chapter 5, page 88-89

[danc]

Hey Tommy,

I have a question about the example question used in the section "Quantity B is an Unknown Value" on pages 88-89 in the QC book, fourth edition. The figure is a circle with center O and two triangles inside the circle. The explanation shows how to find the answer even without exact values for angles or lines or arcs.

My question is, how can the given information be true? The two given statements seem contradictory to me. If O is the center of the circle, then OQ, OR, OS and OP are radii and are all equal in length, as the book states. But doesn't that mean both triangles are isosceles because they each have two sides that are equal? And if they are isosceles, the opposite angles are also equal, which means angles OPS, OSP, ORQ and OQR should all be equal. And if those four angles are equal, the remaining two angles inside the triangles, QOR and POS, should equal each other, but the given info is that one is greater than the other. Am I missing something here?

[tommywallach]

Hey Dan,

You're absolutely right that all these triangles are isosceles, but that doesn't mean we know the angles of the two equal angles. If you just look at these triangles, you can see that the angles are clearly changing. Basically, the more widely spaced the other two points of the triangle are (the two points that are NOT at the center of the circle), the LARGER the angle at the center gets, and the SMALLER the other two angles get.

I think you're mistaking the rule that equal angles in ONE triangle must be opposite equal sides with the idea that ANY equal angles in ANY triangles (Even two separate triangles) should be opposite sides of the same length. But we know this isn't true, from similar triangles. Similar triangles have the same angles but different side lengths. By the same token, we can have the same side lengths but radically different angles in two different isosceles triangles.

-t

[danc]

Ah, right, of course. Thanks for clarifying.

[tommywallach]

No problem! : )