SAT官方每日一题数学（9.3）

Mathematics>Standard Multiple Choice

Note: Figure not drawn to scale.

The sector of a circle shown has center at point O. The radius of the circle has length 1.4 and arc AC has length 2.1. Line segment OB is perpendicular to chord A C and bisects angle AOC. What is the length, l, of chord AC?

Select an Answer:

A:2.8sin(0.75)

B:2.8cos(0.75)

C:1.4sin(1.5)

D:1.4cos(1.5)

Answer A

EXPLANATION

Choice A is correct.

The formula for circumference of a circle is C=2πr, where C is the circumference and r is the radius. For this circle that this sector comes from, the circumference is C=2π(1.4)which is equivalent to C=2.8π.

Because the arc length and circumference are known, a proportion can be used to find the measure of the central angle.

Therefore, angle AOC has measure 1.5 radians.

Because stack OB with bar on top bisects angle AOC, angle BOC has half the radian measure of angle AOC, which is begin display style fraction numerator .

Because triangle OBC is a right triangle, the sine of the measure of angle BOC can be used to find the length of stack BC with bar on top. The sine of an acute angle in a right triangle is equal to the length of the opposite side divided by the length of the hypotenuse:

Line segments AO and CO are congruent because they are both radii. Right triangles OBA and OBC share leg OB, so triangles OBA and OBC are congruent because they have congruent hypotenuses and legs. Therefore, stack AB with bar on top and stack BC with bar on top have the same length, namely 1.4sin(0.75).

Therefore, stack AC has length 2(1.4sin(0.75))=2.8sin(0.75)

难度系数：简单

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