Question

Lines

AB
and
AC
tangent to circle k(O) at B and C respectively. Find BC, if m∠OAB=30°, and AB=5 cm.

Answer 1

5 cm

Explanation:

If AB is tangent to the circle k(O), then radius OB is perpendicular to segment AB.

If BC is tangent to the circle k(O), then radius OC is perpendicular to segment AC.

Consider two right triangles ABO and ACO. In these triangles:

• AO is common hypotenuse;
• ∠OBA=∠OCA=90°, because AB⊥OB, AC⊥OC;
• OB=OC as radii of the circle k(O).

By HL theorem, triangles ABO and ACO are congruent. Then

• ∠OAB=∠OAC=30°;
• AC=AB=5 cm.

Hence, ∠BAC=∠OAB+∠OAC=30°+30°=60°.

Consider triangle ABC, this triangle is isosceles triangle. In isosceles triangles angles adjacent to the base are congruent, thus

∠CBA=∠BCA=1/2(180°-60°)=60°.

Therefore, triangle ABC is an equilateral triangle, so BC=AB=AC=5 cm.