2022 AMC 12B Problem 14

Problem 14 of 25IntermediateGeometry

The graph of

y = x^2 + 2x - 15

intersects the

x

-axis at points

A

and

C

and the

y

-axis at point

B.

What is

\tan(\angle ABC)?

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Solution

Factoring,

x^2 + 2x - 15 = (x+5)(x-3),

so

A = (-5, 0)

and

C = (3, 0),

and the

y

-intercept is

B = (0, -15).

Then

\vec{BA} = (-5, 15)

and

\vec{BC} = (3, 15).

Using the cross and dot products,

\begin{gathered} \tan(\angle ABC) = \scriptsize \dfrac{|(-5)(15) - (15)(3)|}{(-5)(3) + (15)(15)} \\ = \dfrac{120}{210} \\ = \dfrac47. \end{gathered}

Thus, the correct answer is E.

More practice

Concepts: coordinate geometry · vector · trigonometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.