Geometry Tutoring Topics-Example- 2

Geometry Tutoring Topics –Circles and Figures in Circles 2 (cont. Chords)

 

Finding the Length of a Chord 

 

If two radii form an angle in a circle can this angle vary?  Let’s look at an example of an inscribed triangle that makes an isosceles triangle because the radii are the same.  Will this triangle be an equilateral triangle as well?

 

tutoring-geometry-topics

Problem.

Find the length of the chord with a radius of 2 m that has a central angle of 90 deg.

 

This problem assumes that we can use the Length of a Chord Formula

Geometry tutoring topics Length-of-a-Chord/lesson/Length-of-a-Chord-TRIG/

You can solve this problem without it.

If the triangle has 90 degree angle Pythagorean theorem can be use to find the hypotenuse, which is the chord in this case.

 

 

In figure, AB is a diameter of the Circle O, CD is a chord equal to the radius of the circle. AC and BD when extended intersect at a point E.

Prove that angle AEB =60 deg.

 

Solution 1:

  1. Finding the starting Point. (Shortest Solution)

Using the arcs:We know that a central angle spans the arc equal to the length of the central angle. Arc CD = 60. Two secants making an external angle. Using the formula for such an angle we find.

½*(180-60) = ½*(120)= 60 deg.

 

exterior angle theorem

Solution 2:

  1. Finding the starting Point.

Using Triangles:  OCD is equilateral, OD=OC.. radii.

The angles opposite equal sides are equal, -> angle OCD = angle CDO.

Using calculation to find those angles( (180-60 )/2= 60.

Angle ACB is inscribed in a semicircle it must be 90 deg.  Angle ECB makes a linear pair with ACB, 180-90, it is also 90 deg. Angle CBD is half the arc that it is spanning CD= 60 degrees, CBD = ½ * 60 = 30. Find the left over angle of the triangle BCE = 180 – 90 – 30, it is 60.  This is again a 90, 60, 30 triangle.

 

 

Extra Problem Analysis:

Always start with the givens.  Let’s examine what is given. CD chord is equal to the radius of the circle, which is not true for every circle but it is true for this one   à    because the central angle is 60 degree the apex(top) angle in the isosceles triangle. This can be deduced.

angle C =60 degrees , angle D = 60 degrees, angle O = 60 degrees therefore triangle OCD is an equilateral triangle.

 

Problem 2.

Solution 1:

Solution 2:

Suppose you forgot the cosine formula to solve the problem, or you did not have google by your side to help you remember it. Here you can drop a perpendicular and use the 90, 60, 30 triangle to solve this problem. This means X the side of the triangle is equal to2 times 2.5 times square root of 3

= 5 times square root of 3.

Area of a triangle = ½*(BASE ) *(HEIGHT)

Area of a triangle = ½*( X= 5√3) * ( 5+ 2.5)

Area of a triangle = ½*(5√3) * ( 7.5)= approx. 32.47

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