Question 428·Easy·Math

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Points $A$ , $B$ , and $C$ are shown on the $xy$ -plane. Segment $\overline{AB}$ lies on the positive $x$ -axis, and $\angle ABC$ is a right angle. If $\theta$ is the angle at point $A$ between $\overline{AB}$ and $\overline{AC}$ , what is the value of $\tan\theta$ ?

In a coordinate right triangle, you can read the horizontal and vertical leg lengths from the points on the grid. For an angle measured from the horizontal leg, use tangent: opposite divided by adjacent. Whole-number lengths make this a quick calculation without Desmos. Keep the marked angle in view, or you may reverse the two legs.

Hints

Hint 1
The legs are the two sides that meet at the right angle. Read the horizontal distance from $A$ to $B$ and the vertical distance from $B$ to $C$ to find their lengths.
Hint 2
From angle $A$ , the opposite leg is across the triangle, and the adjacent leg touches $A$ . Tangent is opposite over adjacent. Which leg belongs on top of that fraction?

Step-by-step

Read the legs and use tangent
Step 1
Find the two leg lengths
No Desmos needed. The coordinates give both leg lengths directly. The sides meeting at the right angle $B$ are the legs. From $A=(0,0)$ to $B=(9,0)$ , the horizontal leg has length $AB=9-0=9$ . From $B=(9,0)$ to $C=(9,12)$ , the vertical leg has length $BC=12-0=12$ .
Step 2
Put the legs in tangent's order
Stand at angle $A$ . The vertical leg $BC$ is opposite that angle; the horizontal leg $AB$ touches it, so it is adjacent. The slanted side $AC$ is the hypotenuse, across from the right angle, and tangent does not use it. Tangent is opposite over adjacent, measured from the marked angle, so:
$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AB}=\frac{12}{9}.$
Step 3
Reduce the fraction
Divide the top and bottom by $3$ :
$\tan\theta=\frac{12\div3}{9\div3}=\frac{4}{3}.$
So the tangent of the angle at $A$ is $\frac{4}{3}$ . Choice B.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse sine, cosine, and tangent

Question 428·Easy·Math

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