Circles P and Q are externally tangent at T, the radius of P is 12 and the radius of Q is 2. PTQ¯ extended intersects Q at B. A lies on circle P such that AP¯⊥PB¯. R lies on AB¯ such that RT¯⊥PB¯, and S is the intersection of circle Q and AB¯. The length of RS¯ can be written in simplest form as mn. What is m+n?

Question

Circles P and Q are externally tangent at T, the radius of P is 12 and the radius of Q is 2. PTQ¯ extended intersects Q at B. A lies on circle P such that AP¯⊥PB¯. R lies on AB¯ such that RT¯⊥PB¯, and S is the intersection of circle Q and AB¯. The length of RS¯ can be written in simplest form as mn. What is m+n?

Answer 1

To find the length of RS¯, we need to carefully analyze the given information and use some geometric properties.

Let's start by drawing a diagram to better understand the geometry described in the problem.

1. Draw two circles, P and Q, with radii 12 and 2 respectively. Place them so that they are externally tangent at point T.

2. Draw the line extension of PTQ¯. This line intersects circle Q at point B.

3. Draw a line perpendicular from point A on circle P to line PB¯.

4. Next, draw point R on line AB¯, such that RT¯ is perpendicular to PB¯.

5. Finally, draw point S as the intersection of circle Q and line AB¯.

Now that we have a clear picture of the situation, let's proceed with solving the problem.

Since PT is perpendicular to AQ (tangent line perpendicular to radius at the point of tangency), angle PTB is a right angle.

Angle APB is also a right angle because AP¯ is perpendicular to PB¯.

Triangles APB and BRT are similar (by AA similarity) since they both have a right angle and share angle PBR and APT.

Since APB and BRT are similar, we can set up the following proportions:

AB/RT = PB/AP

Applying the lengths we know:

AB/RT = 2/12

Simplifying this ratio, we get:

AB/RT = 1/6

Now, using similar triangles, we can also set up the following proportion:

RS/AB = RT/PB

Plugging in the known lengths:

RS/AB = RT/2

Since AB/RT = 1/6 (from the previous proportion), we can substitute this value in the above proportion:

RS/(1/6) = RT/2

Simplifying, we get:

RS = (RT/2) * (1/6)

RS = RT/12

So, the length of RS¯ is RT/12.

Now, let's determine the length of RT¯. Let's consider the right triangle BRT.

By the Pythagorean theorem, we have:

RT^2 + BT^2 = BR^2

Since BT is the radius of Q and given as 2, and since BR is equal to AB (because RT is perpendicular to PB), we can rewrite the equation as:

RT^2 + 2^2 = AB^2

Simplifying further:

RT^2 + 4 = AB^2

Since AB/RT = 1/6, we can substitute 6RT for AB:

RT^2 + 4 = (6RT)^2

Equating the terms, we have:

RT^2 + 4 = 36RT^2

35RT^2 = 4

RT^2 = 4/35

Taking the square root:

RT = √(4/35) = 2/√35

Therefore, the length of RS¯ is RT/12 = (2/√35) / 12 = 2/(12√35) = √35 / 210.

From this, we can tell that m = √35 and n = 210. Thus, m + n = √35 + 210.