In the below figure, EB is tangent to the circle with center $C$ at the point B. If EC is equal to the diameter of the circle, find the measure of angle BAD in degrees.

Question

In the below figure, EB is tangent to the circle with center $C$ at the point B. If EC is equal to the diameter of the circle, find the measure of angle BAD in degrees.

and here is the ugly LaTeX
[asy]
pair C = (0,0), B = dir(110), D = dir(170), E2 = 2 * D, A = dir (305);
draw(unitcircle);
draw(C--E2--B--C^^D--A--B);
dot(A^^B^^C^^D^^E2);
label("$A$",A,SE);
label("$B$",B,NW);
label("$C$",C,S);
label("$D$",D,SW);
label("$E$",E2,W);
[/asy]

Answer 1

Your post is undecipherable .

I can draw the circle, its centre O, and the tangent EB.
Where is A and D in angle BAD ???
I can find the central angle BCE by
cos BCE = r/(2r) = 1/2
so angle BCE = 60° , which makes angle BEC = 30°

Does that help?

Answer 2

thanks that helped a lot

Answer 3

Let's analyze the given figure step by step to find the measure of angle BAD.

Step 1: Since EB is tangent to the circle at point B, we know that angle ECB is 90 degrees. This is because a tangent line to a circle is always perpendicular to the radius drawn at the point of tangency.

Step 2: Given that EC is equal to the diameter of the circle, we can conclude that angle EBC is also 90 degrees. This is because the diameter of the circle passes through the center, and angles formed by a diameter and a chord are always 90 degrees.

Step 3: Since the sum of the angles in a triangle is 180 degrees, we can find angle CEB by subtracting the measures of angles ECB and EBC from 180 degrees. Therefore, angle CEB = 180 - 90 - 90 = 0 degrees.

Step 4: Considering that angle CEB is 0 degrees, we can conclude that angle AEB is 180 degrees. This is because the sum of the measures of angles around a point is always 360 degrees, and angle CEB is a straight angle.

Step 5: Finally, we can find angle BAD by subtracting the measures of angles AEB and EBA from 180 degrees. Therefore, angle BAD = 180 - 180 - 90 = -90 degrees.

Step 6: However, angles are always measured between 0 and 180 degrees, so we need to find an equivalent positive angle. Adding 180 degrees to -90 degrees will give us an equivalent angle that is between 0 and 180 degrees:

Angle BAD = -90 + 180 = 90 degrees.

Therefore, the measure of angle BAD in degrees is 90 degrees.

Answer 4

To find the measure of angle BAD in degrees, we need to determine the relationship between angle BAD and the circle.

Since EB is tangent to the circle at point B, we know that the tangent line is perpendicular to the radius at the point of tangency. Therefore, angle EBC is a right angle.

Given that EC is equal to the diameter of the circle, we can infer that triangle BCE is an isosceles right triangle.

Since angle EBC is a right angle, the other two angles in triangle BCE must be acute and equal. Let's call these angles x.

Since triangle BCE is isosceles, the other two sides (BE and CE) are equal in length. Therefore, angle BEC is also equal to x degrees.

Since the sum of the angles in a triangle is 180 degrees, we can write the equation:

x + x + 90 = 180

Simplifying the equation:

2x + 90 = 180

Subtracting 90 from both sides:

2x = 90

Dividing both sides by 2:

x = 45

So, angle EBC and angle BEC are both equal to 45 degrees.

Now, we can find the measure of angle BAD. Since angle BAD is an exterior angle of triangle BCE, it is equal to the sum of the two remote interior angles.

Therefore, angle BAD = angle EBC + angle BEC = 45 + 45 = 90 degrees.

Hence, the measure of angle BAD is 90 degrees.