Question 516·Medium·Math

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In right triangle $\triangle ABC$ , $\angle C$ is a right angle, $\angle A$ measures $35^{\circ}$ , and side $AC$ (the leg adjacent to $\angle A$ ) has length $12$ units.

Which expression represents the length of the hypotenuse $AB$ ?

In a right triangle, the cosine ratio connects the leg beside an acute angle to the hypotenuse. If the known leg is adjacent and the unknown side is the hypotenuse, write cosine as adjacent over hypotenuse before rearranging. When the question asks for an expression rather than a decimal, you don't need to evaluate the angle.

Hints

Hint 1
The hypotenuse is always across from the right angle. The adjacent leg touches the angle you're using and isn't the hypotenuse. Which side has each role here?
Hint 2
The cosine ratio is adjacent over hypotenuse. Put the known length on top and the unknown length on the bottom. How can you move the unknown out of the denominator?

Step-by-step

Use the cosine ratio
Step 1
Identify the two sides
No Desmos needed. The question asks for an expression, not a decimal. Since $\angle C$ is right, $AB$ is across from it, so $AB$ is the hypotenuse, the longest side. From $\angle A$ , $AC=12$ is the adjacent leg: it touches that angle and isn't the hypotenuse.
Step 2
Write the ratio
Use cosine because it relates the two sides you know and need: cosine is adjacent over hypotenuse. For the $35^\circ$ angle, write:
$\cos 35^\circ=\frac{12}{AB}.$
Step 3
Move the hypotenuse out of the denominator
Multiply both sides by $AB$ so the unknown leaves the denominator:
$AB\cos 35^\circ=12.$
Step 4
Isolate the hypotenuse
Divide by $\cos 35^\circ$ to get the length of $AB$ : $AB=\frac{12}{\cos 35^\circ}$ . So the hypotenuse is $\frac{12}{\cos 35^\circ}$ units. Choice D.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse sine, cosine, and tangent

Question 516·Medium·Math

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Step-by-step

Lessons that teach this

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Step-by-step

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