Points A, B, and C are on a circle such that angle ABC=45 degrees, AB=6, and AC=8. Find the area of the circle.
How about this way:
Since AC supports a 45 degree angle at the circle, AC must support a 90 degree angle at the centre.
let the radius be r
r^2 + r^2 = 8^2
2r^2 = 64
r^2 = 32
Area = pi r^2 = 32 pi or appr 100.53
Notice just as Damon pointed out, we don't even need to know that AB = 6
AB = 6
AC = 8
all I need to know actually is AC = 8
so
circumference = 8 (360/45)
= 64 = 2 pi r
r = 32/pi
pi r^2 = pi (1024/pi^2) = 326
whatevers, miles, astronomical units, thumbs (I do physics not math)
Hmm 326 is not right... I'm also confused on "circumference = 8 (360/45) = 64 = 2 pi r" can you explain that a bit?
Oh I see! Yes, your answer is correct Reiny. Thanks!
To find the area of the circle, we need to find the radius of the circle first.
Given that angle ABC is 45 degrees, we can conclude that angle ACB is also 45 degrees. This is because angles subtended by an arc on the circumference of a circle from a common chord are equal.
Now, let's use the Law of Cosines to find the length of BC, which is the diameter of the circle.
The Law of Cosines states that in any triangle ABC, we have:
c^2 = a^2 + b^2 - 2ab * cosC
Here, a = AC = 8, b = AB = 6, and C = angle ACB = 45 degrees. Plugging these values into the formula, we get:
BC^2 = 8^2 + 6^2 - 2 * 8 * 6 * cos(45)
BC^2 = 100
BC = 10
Since BC is the diameter of the circle, we can conclude that the radius (r) of the circle is half of BC. Therefore, r = 10 / 2 = 5.
Now that we have the radius, we can calculate the area of the circle using the formula: Area = π * r^2.
Area = π * 5^2 = 25π.
So, the area of the circle is 25π square units.