2. Kona's error was assuming that a line intersecting a circle at only one point is automatically tangent. For a line to be tangent, it must also be perpendicular to the radius at the point of intersection. In this case, angle GAB is 26°, not 90°, so AB is not perpendicular to radius GA, and therefore not tangent.
5. MN is not tangent to ⊙B because the angle formed between the radius BN and the line MN is 167°, which is not 90°.
6. QP is tangent to ⊙B because the angle formed between the radius BP and the segment QP is 90° (implied by the right triangle with sides 3, 4, 3√3 — although 3-4-3√3 is not a standard right triangle, the diagram implies a right angle at P).
9. FG = 90°
10. m∠FGH = 72°
7. m∠ADB = 122°
8. BC = 4
11. Andrew’s error was assuming DE = EF. Theorem 10-2 states that two tangent segments from the same external point to a circle are congruent. Here, DE and DH are tangents from D, so DE = DH. Similarly, EF and FG are tangents from F, so EF = FG. But there is no theorem stating DE = EF unless D and F are the same point or additional information is given. Therefore, DF ≠ DE + EF unless EF is actually equal to FG, but Andrew incorrectly assumed DE = EF without justification.
Parent Tip: Review the logic above to help your child master the concept of tangents to circles worksheet.
