2021 AMC 12B Fall Problem 22

Attempt Problem 22 of the 2021 AMC 12B Fall below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Fall solutions, or check the answer key.

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22.

Right triangle

ABC

has side lengths

BC = 6,

AC = 8,

and

AB = 10.

A circle centered at

O

is tangent to line

BC

at

B

and passes through

A.

A circle centered at

P

is tangent to line

AC

at

A

and passes through

B.

What is

OP?

\dfrac{23}{8}

\dfrac{29}{10}

\dfrac{35}{12}

\dfrac{73}{25}

3

Answer: C

Concepts:coordinate geometry tangent line distance formula

Difficulty rating: 2490

Small hint:

Place

C

at the origin with legs along the axes

Big hint:

A tangent point’s radius is perpendicular to the tangent line, so

O

lies directly above

B

and

P

directly across from

A

Solution:

Place

C = (0, 0),

B = (6, 0),

and

A = (0, 8),

so the right angle is at

C.

Circle

O

is tangent to line

BC

(the

x

-axis) at

B,

so

O = (6, k).

Setting

OA = OB

gives

36 + (k - 8)^2 = k^2,

so

k = \tfrac{25}{4}

and

O = \left(6, \tfrac{25}{4}\right).

Circle

P

is tangent to line

AC

(the

y

-axis) at

A,

so

P = (h, 8).

Setting

PB = PA

gives

(h - 6)^2 + 64 = h^2,

so

h = \tfrac{25}{3}

and

P = \left(\tfrac{25}{3}, 8\right).

Then

OP = \sqrt{\left(\tfrac73\right)^2 + \left(\tfrac74\right)^2}

= 7\sqrt{\tfrac{25}{144}}

= \dfrac{35}{12}.

Thus, the correct answer is C.

Problem 22 in other years