Tutoring Geometry Topics –Circles and Figures in Circles 1.
Finding the starting point.
The right starting point can lead to a faster, an easier solution. It is essential to tutor with intent on making a student a better problem solver.
a. What are the givens?
Examining all the given information to find the right place to start is crucial. In this example we are given the radii of the circle. àWhere can this information lead us?
Remember the more pertinent the information that can lead us to the solution the more likely it is a better starting point for an efficient solution.
Knowing the Radii of a circle->
I can Find the Circumference , Area of circle
Knowing that the segment BC is tangent
to a circle at B -> I can Find that a tangent to a circle makes a right ..90 degree angle with a circle.
OB=OA=AC=6 -> I can Find the Hypotenuse of the triangle OC=OA+AC and use the Pythagorean Formula.
Which entry point should be used.. Lets look at the question: it says that we need to find the area of the shaded region. To do this we need to explore and find the areas of the figures that include this shaded area, and since all of the choices have subtraction in them (this gives us a hint that we need to find the larger area and then subtract).
How do we find the larger area and what larger area should we use? You guessed it right it is the triangle. Since the shaded area is inside the triangle we need to find this area and then subtract the shaded area from the triangle.
What information do we know about this triangle? -Yes. It is a right triangle. We know this from the fact that tangent to a radius makes a (right) 90 degree angle.
What information do we know that will help us find the area of this triangle (that is what we are aiming for)?-
Let us Work backwards from the choices.
Notice the choices: All of them have a √3 in them.
By association, or using Pythagorean theorem
we can associate a 90, 60, 30 triangle since angle ratios are 2a, a√3, a.
Good idea, this would mean that one of the angles is 60. It looks like it could be the central angle. But can we back this information up with formal Geometry theorems, rules?
This picture clearly shows the radii of a circle making a triangle with the third line being a chord.
Similar example can be found here
Angle C = 30. To find the area of this triangle (that is what we are aiming for) we need two sides base and height. Since this is a right triangle the base and height make a 90 degree angle BC – height BO- base.
Segment BO = 6
Segment BC = ? Pythagorean Triplets or Formula is always used whenever there is a right triangle.
According to the right triangle rules if x is opposite of 30, x multiplied by √3 = 6 multiplied by√3.
Now we can find the area of the triangle (finally)
Area of right triangle
(BASE x HEIGHT)/2 = (6X 6√3)/2 = 3 X 6√3 = 18√3.
What is left to find?
To find the answer (the shaded region) we need to subtract the area of the non-shaded region from the whole region (the triangle).
We already know the area of the triangle= 18√3.
Now we need to find the area of the sector with radius of 6 and central angle of 60 deg. Here you may use the formula to do so.
If in case you forget the formula, this is the logical explanation. Since all of the circle is 360 deg, and the area of the whole circle is pi times r squared.