Question

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.

 

Finding Segment AD:

 

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))

 

Answer:

 

The length of PA is 18√3.