Unformatted text preview: ACB is a right angle (see Figure 1.1.3(a)). In other words, the triangle △ ABC is a right triangle. A B C O (a) A B C O α α β β (b) Figure 1.1.3 Thales’ Theorem: ∠ ACB = 90 ◦ To prove this, let O be the center of the circle and draw the line segment OC , as in Figure 1.1.3(b). Let α = ∠ BAC and β = ∠ ABC . Since AB is a diameter of the circle, OA and OC have the same length (namely, the circle’s radius). This means that △ OAC is an isosceles triangle, and so ∠ OCA = ∠ OAC = α . Likewise, △ OBC is an isosceles triangle and ∠ OCB = ∠ OBC = β . So we see that ∠ ACB = α + β . And since the angles of △ ABC must add up to 180 ◦ , we see that 180 ◦ = α + ( α + β ) + β = 2( α + β ), so α + β = 90 ◦ . Thus, ∠ ACB = 90 ◦ . QED...
View
Full Document
 Fall '11
 Dr.Cheun
 Calculus, Angles, Right triangle, triangle

Click to edit the document details