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In the diagram, AD is a diameter of the circle, angle BAC = 60, BE is perpendicular to AC, and EC = 3.  Find the length of chord BD.

 

 Dec 31, 2023
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Isosceles Triangle ABC: Since AD is a diameter and angle BAC is 60°, triangle ABC is equilateral. Therefore, AB = AC.

 

Right Triangle BCE: Triangle BCE is a right triangle with BE perpendicular to AC.

 

We are given EC = 3, and since AC = AB (from step 1), we can use the Pythagorean theorem to find BC. BC^2 = AC^2 - EC^2 = AB^2 - EC^2.

 

Finding AB: Let x be the length of AB (and AC). Substituting from step 2 and rearranging, we get x^2 = 3^2 + x^2, which simplifies to x^2 = 9. Solving for x, we find AB = AC = 3√3.

 

Finding BD: Now that we know AB = 3√3, we can use the properties of triangles again. Since AD is a diameter, right triangle ABD has a right angle at B. Angle ADB = 180° - 90° - 60° = 30°.

 

Right Triangle ABD: Applying the sine function to triangle ABD, we get BD/AB = sin(30°). Substituting AB = 3√3, we get BD = AB * sin(30°) = 3√3 * (1/2) = 3 * (√3/2).

 

Final Answer: Therefore, the length of chord BD is 3 * (√3/2) = 3√6/2.

 Dec 31, 2023

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