The question is in Vietnamese and involves geometry, specifically dealing with a circle and tangents. Here is the translation and solution to the first three parts of the question:
Question 35: Given a circle (O) and a point A outside the circle. Draw the tangents AB and AC from point A to the circle, touching the circle at points B and C respectively.
a) Prove that OA is perpendicular to BC.
b) Draw the diameter CD. Prove that BD is parallel to AO.
c) Calculate the lengths of the sides of triangle ABC, given OB = 2 cm and OA = 4 cm.
a) Prove that OA is perpendicular to BC.
To prove that OA is perpendicular to BC, we can use the property of tangents to a circle from an external point.
- Tangents from a common external point are equal in length. Therefore, AB = AC.
- Since AB and AC are tangents from point A to the circle, they make equal angles with the line segment OA.
- Let the center of the circle be O. Then, OB and OC are radii of the circle.
- Since AB and AC are tangents, they are perpendicular to the radii at the points of tangency. Therefore, OB ⊥ AB and OC ⊥ AC.
- Triangles OAB and OAC are right triangles with OB and OC as the legs perpendicular to the tangents.
- Since AB = AC, triangles OAB and OAC are congruent by the hypotenuse-leg (HL) theorem.
- Therefore, angles OAB and OAC are equal.
- Since the sum of angles OAB and OAC is 180° (straight line), each angle is 90°.
- Thus, OA is the perpendicular bisector of BC, meaning OA ⊥ BC.
b) Draw the diameter CD. Prove that BD is parallel to AO.
To prove that BD is parallel to AO:
- Draw the diameter CD of the circle.
- Since CD is a diameter, it passes through the center O of the circle.
- The line segment BD is a chord of the circle.
- Since AB and AC are tangents from point A, and AB = AC, triangle ABC is isosceles with AB = AC.
- The line segment AO is the angle bisector of angle BAC.
- Since CD is a diameter, angle BCD is a right angle (90°) because the angle subtended by a diameter in a semicircle is a right angle.
- Therefore, angle BCD = 90°.
- Since AO is the angle bisector of angle BAC and angle BCD = 90°, BD is parallel to AO by the corresponding angles postulate.
c) Calculate the lengths of the sides of triangle ABC, given OB = 2 cm and OA = 4 cm.
To calculate the lengths of the sides of triangle ABC:
- Given OB = 2 cm (radius of the circle) and OA = 4 cm.
- Since AB is a tangent to the circle at point B, OB ⊥ AB.
- Triangle OAB is a right triangle with OB as one leg and OA as the hypotenuse.
- Using the Pythagorean theorem in triangle OAB:
\[
AB = \sqrt{OA^2 - OB^2} = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3} \text{ cm}
\]
- Similarly, AC = 2√3 cm (since AB = AC).
Therefore, the lengths of the sides of triangle ABC are:
- AB = 2√3 cm
- AC = 2√3 cm
- BC = 2 * AB = 2 * 2√3 = 4√3 cm (since AB = AC and triangle ABC is isosceles).
In summary:
a) OA is perpendicular to BC.
b) BD is parallel to AO.
c) The lengths of the sides of triangle ABC are AB = 2√3 cm, AC = 2√3 cm, and BC = 4√3 cm.
Answered
