Question

Let AB be a diameter of a circle with diameter 2r. Let P and Q be points on one of the semicircular arcs determined by AB, such that P is the midpoint of the semicircle and AP = r. Point C lies on the other semicircular arc. Let AC and BC be the lengths of the line segments whose endpoints are the intersections of diameter AB with the chords PC and QC. The largest possible value of AC + BC can be written in the form a√b, where a, b, and are positive integers, and a is not divisible by the square of any prime. Find a.

A) 2√2
B) 3√2
C) 4√2
D) 5√2

Answer 1

To find the largest possible value of AC + BC, we can use the Pythagorean theorem to find the lengths of AC and BC and then calculate their sum. The largest possible value of AC + BC can be written as a√2, where a is a positive integer and is not divisible by the square of any prime.

Step-by-step explanation:

To find the largest possible value of AC + BC, we need to analyze the given information.

From the information provided, we know that AB is a diameter of the circle, AC and BC are perpendicular to AB and intersect the chords PC and QC respectively.

Let's break down the problem step by step:

1. First, we note that AP is equal to r, so PC is an altitude of the triangle APC.
2. Since P is the midpoint of the semicircle, the measure of angle PAC is 90 degrees.
3. This means that triangle APC is a right triangle with AC as the hypotenuse and PC as an altitude.
4. Similarly, BQ is equal to r, so QC is an altitude of the triangle BQC.
5. Again, the measure of angle QBC is 90 degrees, so triangle BQC is a right triangle with BC as the hypotenuse and QC as an altitude.
6. We can use the Pythagorean theorem to find the lengths of AC and BC:

AC = √(AP^2 + PC^2) = √(r^2 + (2r)^2)

BC = √(BQ^2 + QC^2) = √(r^2 + (2r)^2)

Since AC and BC are equal, the largest possible value of AC + BC is 2 * AC = 2 * √(r^2 + (2r)^2).

Given that r is positive, the largest possible value of AC + BC can be written as a√(2) where a is a positive integer and is not divisible by the square of any prime.