PSAT Math : How to find the length of a chord

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #21 : Circles


The circle above has a radius of , and the measure of  is . What is the length of chord ?

Possible Answers:

Correct answer:


To solve a chord problem, draw right triangles using the chord, the radii, and a line connecting the center of the circle to the chord at a right angle. 


Now, the chord is split into two equal pieces, and angle AOB is bisected. Instead of one 120 degree angle, you now have two 30-60-90 triangles. 30-60-90 triangles are characterized by having sides in the following ratio:Triangle306090-3

So, to find the length of the chord, first find the length of each half. Because the triangles in your circle are similar to the 30-60-90 triangle above, you can set up a proportion. The hypotenuse of our triangle is 6 (the radius of the circle) so it is set over 2 (the hypotenuse of our model 30-60-90 triangle). Half of the chord of the circle is the leg of the triangle that is across from the 60 degree angle (120/2), so it corresponds to the  side of the model triangle. 


Because x is equal to half of the chord, the answer is .

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