In the figure above, triangle ABC is inscribed in a circle whose centre O has the x- and y-coordinates as (0,0). If the x- and y- coordinates of point A are (-4,0) and andang;BAC = 30, what is the area, in square units, of triangle AOB?a)2andradic;3b)4andradic;3c)8andradic;3d)16andradic;3e)24andradic;3Correct answer is option 'B'. Can you explain this answer?

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In the figure above, triangle ABC is inscribed in a circle whose centre O has the x- and y-coordinates as (0,0). If the x- and y- coordinates of point A are (-4,0) and ∠BAC = 30^∘, what is the area, in square units, of triangle AOB?

a)
2√3
b)
4√3
c)
8√3
d)
16√3
e)
24√3

Correct answer is option 'B'. Can you explain this answer?

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Test: Circles- 2

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In the figure above, triangle ABC is inscribed in a circle whose centr...

Given:

O is the center of the circle and A(-4,0) is a point on the circle

So, radius of the circle, R = Distance between points O and A = 4 units
Also, AC is the diameter of the circle.

So, ∠ABC is an angle in the semicircle

So, ∠ABC = 90^o
In right ΔABC,
- ∠BAC = 30∘

So, by Angle Sum Property, ∠ACB = 60^o

To find:

Area of (ΔAOB)

Approach:

Let’s drop a perpendicular BD from point B on AC.

Area triangle AOB = ½ * AO * BD

We’ve inferred the length of AO (= radius of the circle) in the Given section above. So, to answer the question, we need to find the length of BD
In ΔOBC, sides OB and OC are equal because both are the radii of the circle.

So, this is an isosceles triangle.

Since we already know ∠OCB=60, therefore ∠OBC will also be equal to 60.
Which finally helps us in inferring that ∠BOC = 60^o
Hence we can conclude that OBC is an equilateral triangle.
The ΔODB, is a 30-60-90 Triangle and we know the measure of radius OB = 4 units. Thus using the side property of 30-60-90 Triangle we can write –

OD: BD: OB = 1: √3: 2

BD: OB = √3: 2
Thus BD = 2√3
Thus area of triangle AOB = ½ * AO * BD = ½ * 4 * 2√3 = 4√3
Looking at the answer choices, we see that the correct answer is Option B.

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In the figure above, triangle ABC is inscribed in a circle whose centre O has the x- and y-coordinates as (0,0). If the x- and y- coordinates of point A are (-4,0) and ∠BAC = 30∘, what is the area, in square units, of triangle AOB?a)2√3b)4√3c)8√3d)16√3e)24√3Correct answer is option 'B'. Can you explain this answer?

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