SOLUTION: The figure shows Traingle ABC inscribed in a circle with centre O. If Angle CBO is twice of Angle CAO and angle BAO is one and a half times of angle CBO, find angle CAO. This is

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Question 1027673: The figure shows Traingle ABC inscribed in a circle with centre O. If Angle CBO is twice of Angle CAO and angle BAO is one and a half times of angle CBO, find angle CAO.
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Answer by Theo(13342) (Show Source):
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there are a couple of ways to solve this.

one finds the central angle of each triangle formed.

the other finds the number of degrees of the intercepted arcs on the circle.

the sum of the central angles of the circle is equal to 360 degrees.

an inscribed angle of the circle is equal to 1/2 the degrees of the intercepted arc on the circle.

the sum of the degrees of the intercepted arcs on the circle is equal to 360 degrees.

both these methods yield the same answer.

that answer is that angle CAO = 15 degrees and angle CBO = 30 degrees and angle BAO = 45 degrees.

you are given that angle CBO is twice angle CAO.

if we let x = angle CAO, then angle CBO must be equal to 2x.

you are also given that BAO is equal to 1.5 times angle CBO.

since angle CBO is equal to 2x, this mean that angle BAO must be equal to 3x.

you have:

angle CAO = x
angle CBO = 2x
angle BAO = 3x.

these angles are the base angles of isosceles triangles.

the isosceles triangles are triangle CAO, CBO, and BAO.

since the base angles of isosceles triangles are equal, this means that:

angle ACO = x
angle BCO = 2x
angle ABO = 3x

since the sum of the angles of any triangle is equal to 180 degrees, this means that:

angle COA = 180 - 2x
angle COB = 180 - 4x
angle BOA = 180 - 6x

the sum of these 3 central angles of the circle must be equal to 360 degrees.

this means that 540 - 12x = 360 degrees.

subtract 360 from both sides of this equation and add 12x to both sides of this equation to get 540 - 360 = 12x

solve for x to get x = (540 - 360) / 12 = 15 degrees.

this means that:

angle CAO = x = 15 degrees.
angle CBO = 2x = 30 degrees.
angle BAO = 3x = 45 degrees.

the central angles become:

angle COA = 180 - 2x = 180 - 30 = 150
angle COB = 180 - 4x = 180 - 60 = 120
angle BOA = 180 - 6x = 180 - 90 = 90

the sum of these central angles is equal to 360 degrees, as it should be.

you could also have solved this by extending the radii of the circle to form diameters and then looking at the intercepted arcs.

these intercepted arcs are equal to twice the inscribed angle.

when you do that you get 24 * x = 360 degrees.

when you solve for x, you get x = 360 / 24 = 15 degrees.

it's the same answer.

i drew the both diagrams for you to see.

the first is using the central angle to find.

the second is using the intercepted arc angle to find x.