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Right triangles and trigonometry SAT question

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

The figure presents triangle A, B C, where side A, C is horizontal and point B is above side A, C. Point D lies on side A, C, and line segment B D is drawn, forming right angle B D C. Side B C is labeled 12. Angle A, B D is labeled 30 degrees and angle B C D is labeled 60 degrees.

In $triangle A, B C$ above, what is the length of $segment A, D$ ?

ExplanationShow

Choice B is correct. Triangles ADB and CDB are both $30 degree 60 degree 90 degree$ triangles and share $side B D$ . Therefore, triangles ADB and CDB are congruent by the angle-side-angle postulate. Using the properties of $30 degree 60 degree 90 degree$ triangles, the length of $side A, D$ is half the length of hypotenuse $A, B$ . Since the triangles are congruent, $the length of side A, B equals the length of side B C, which equals 12$ . So the length of $side A, D$ is $twelve halves equals 6$ .

Alternate approach: Since angle CBD has a measure of $30 degrees$ , angle ABC must have a measure of $60 degrees$ . It follows that triangle ABC is equilateral, so side AC also has length 12. It also follows that the altitude BD is also a median, and therefore the length of AD is half of the length of AC, which is 6.

Choice A is incorrect. If the length of $side A, D$ were 4, then the length of $side A, B$ would be 8. However, this is incorrect because $side A, B$ is congruent to $side B C$ , which has a length of 12. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives $side A, B$ a length of $12 times the square root of 2$ for choice C and $12 times the square root of 3$ for choice D. However, these results cannot be true because $side A, B$ is congruent to $side B C$ , which has a length of 12.

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