P is a point outside of circle Γ. The tangent from
P to
Γ
touches at A. A line from
P intersects
Γ at
B and
C such
that ∠ACP=120
∘. If
AC=16 and
AP=19, then the
radius of Γ can be written as a root b/c
where a and c are co prime and c is not divisible by square of any prime.Find a+b+c.
To find the radius of circle Γ, we can use the relationship between the lengths of tangents drawn from an external point to a circle. Let's go step by step.
1. Start by drawing the figure with the given information. Place the center of the circle Γ at point O and mark the point P outside the circle. Draw the tangent line from P, touching circle Γ at point A. Draw a line passing through P, intersecting the circle at points B and C, with ∠ACP = 120°. Mark AC = 16 and AP = 19.
2. It is given that the tangent from P touches the circle at point A. This implies that the line segment PA is perpendicular to the tangent line at A. Therefore, triangle PAB is a right triangle with PA as the hypotenuse and AB as the radius of circle Γ.
3. By using Pythagoras' theorem on triangle PAB, we can find AB:
AB² = AP² - BP² (Since PA is the hypotenuse, AP is the known side length, and BP is the unknown side length)
AB² = 19² - (AC/2)² (Since BP = AC/2, as both B and C bisect line segment AC)
AB² = 19² - 8²
AB = √(19² - 8²)
4. Now, we have found the length of AB, which is the radius of circle Γ. However, the answer needs to be in the form of a root b/c, where a and c are co-prime and c is not divisible by the square of any prime.
To simplify the expression, calculate the value of (19² - 8²) and express it as a root b/c:
AB = √(19² - 8²)
= √(361 - 64)
= √297
= √(3 * 99)
= √(3 * 3 * 11)
= 3√11
Therefore, the radius of circle Γ is 3√11. In this case, a = 3, b = 11, and c = 1. (Since 1 is not divisible by the square of any prime) Hence, a + b + c = 3 + 11 + 1 = 15.