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# Question

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Equilateral triangle ABC is inscribed in a circle. Let P be a point on arc BC. Given that PB=18 and PC=7 find PA.

Apr 13, 2024

#1
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Here's how to find the length of PA (the distance between point P and point A) in the given scenario:

Properties of an Equilateral Triangle:

Since ABC is an equilateral triangle, all three sides (AB, AC, and BC) are equal in length.

Triangle Angle Bisection by Altitude:

The altitude drawn from a vertex of an equilateral triangle bisects the opposite side and also creates two 30-60-90 right triangles.

Applying Triangle Properties:

Let D be the midpoint of BC. Since the altitude from A bisects BC, point D coincides with the midpoint of segment PA.

We are given that PB = 18 and PC = 7. Since BC is divided into two segments with a ratio of 18:7, segment BD must have a length of 18 and segment CD a length of 7.

30-60-90 Right Triangle

:

Triangle BDP is a 30-60-90 right triangle because BD is half the hypotenuse of the equilateral triangle (BC), and the altitude from A bisects the base at a 60-degree angle (property of equilateral triangles).

In a 30-60-90 triangle, the shorter leg (opposite the 30-degree angle) is half the length of the hypotenuse. In this case, BD (shorter leg) = 18, so the hypotenuse (BP) is twice that length, which is 36.

Since triangle BDP is 30-60-90, the longer leg (opposite the 60-degree angle) is equal to the shorter leg multiplied by the square root of 3. We know the shorter leg (BD) is 18, so:

AD (longer leg) = BD * √3 = 18 * √3

Finding Segment PA:

Since D is the midpoint of segment PA, then PD = DA = (18 * √3) / 2

Now we can find the total length of PA by adding the lengths of segments PD and DA:

PA = PD + DA = (18 * √3) / 2 + (18 * √3) / 2

PA = 18√3 (We can simplify this further if needed, but the answer accepts sqrt(3))