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Right triangles and trigonometry SAT question

Math · Geometry and Trigonometry · Hard · Answer it here, explanation included.

The figure presents right triangle A, B C. Side A, C is horizontal, and vertex B is directly above vertex C. Angle C is a right angle. The length of side A, B is 26.

Triangle $A, B C$ above is a right triangle, and $the sine of B, equals 5 over 13$ . What is the length of side $B C$ ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle B, or $sine of B$ , is equal to the ratio of the length of side $A, C$ to the length of side $A, B$ . It’s given that the length of side $A, B$ is 26 and that $sine of B equals, 5 over 13$ . Therefore, $the fraction 5 over 13, equals, the fraction A, B over 26$ . Multiplying both sides of this equation by 26 yields $A, C equals 10$ .

By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle ABC is as follows: $26 squared equals, 10 squared, plus B C squared$ , or $676 equals, 100 plus B C squared$ . Subtracting 100 from both sides of $676 equals, 100 plus B C squared$ yields $576 equals, B C squared$ . Taking the square root of both sides of $576 equals, B C squared$ yields $24 equals B C$ .

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