Circle Geometry question

In summary: Since they are equal, BAC must be twice their size. So, BAC must be greater than 90 degrees.In summary, in the given scenario, point B and C are common points on two circles with the same radius. By using basic geometry principles, it can be determined that the measure of angle BAC must be greater than 90 degrees.
  • #1
PsychonautQQ
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In the euclidean plane, point A is on a circle centered at point O and point O is on a circle centered at point A. What is measure of angel BAC?

So I drew a picture, and it seems that BAC is going to definitely be greater than 90 degree's. From there I am confused on what to do next. Anyone have some advice?
 
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  • #3
I assume that B and C are the two common points on the circles.

Now, the radii of both circles are the same (why?). Now remember elementary geometry. Draw the circle around A using a compass. Put the point of the compass at point O and mark the point B (without changing the span of the compass). Now, the distance AO=r and AB=r (they are both on the circle around A). In addition, OB=r (the compass gave the distance). Thus, in the triangle AOB, all sides are equal, an therefore all angles are equal (the size of the angle is left as an exercise for the student).
Now, assume that you did not just mark point B, but let the compass draw a complete circle. In addition to point B you would also have a point C where the circles intersect. Then everything we just said about the triangle AOB is also valid for the triangle AOC. Now, the angle BAC is the sum of the angles BAO and OAC.
 

What is Circle Geometry?

Circle Geometry is a branch of mathematics that deals with the properties and relationships of circles and other curved shapes. It involves the study of angles, chords, tangents, and other geometric concepts related to circles.

What are the basic elements of a circle?

The basic elements of a circle include the center, radius, diameter, circumference, and arc. The center is the point equidistant from all points on the circle, while the radius is the distance from the center to any point on the circle. The diameter is a line segment that passes through the center and connects two points on the circle, and the circumference is the distance around the circle. An arc is a curved line segment that forms a part of the circumference.

What is the relationship between a circle's radius and diameter?

The radius and diameter of a circle are related by the formula: diameter = 2 * radius. This means that the diameter is always twice the length of the radius.

How do you find the area of a circle?

The formula for finding the area of a circle is A = π*r^2, where A is the area and r is the radius. This means that the area of a circle is equal to pi (approximately 3.14) multiplied by the square of the radius.

What are some real-life applications of Circle Geometry?

Circle Geometry has numerous real-life applications, including architecture, engineering, navigation, and art. For example, architects use circle geometry to design curved structures, engineers use it to calculate the strength of circular structures, navigators use it to plot the course of ships and planes, and artists use it to create circular designs and patterns.

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