ShmoopTube

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Secant angles, a la Shmoop.

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Roger Goosebuss has invented a new amphibious vehicle that he would like

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to test for navigational accuracy.

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To do this, he employed two of his new vehicle prototypes and a perfectly circular lake…

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…located a few miles from his manufacturing headquarters.

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To test whether his vehicles are capable of traveling in exactly straight paths…

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…Roger will have the vehicles take off on two routes that form an angle of 45 degrees.

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They will reach the lake at two points, forming an arc of 60 degrees…

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…and then they’ll continue their respective paths across the lake,

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arriving somewhere on the opposite shore.

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If their journey is successful and the vehicles travel in perfectly straight lines…

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…what should be the arc measure between them when they arrive on the far shore of the lake?

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The formula for finding any arc or angle measure involved in a secant angle problem such as this…

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...is secant angle K equals one-half the difference between the measurements of the

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far arc A-B-C and near arc X-Y-Z.

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With this formula, it’s a simple matter of… plugging in.

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The secant angle of 45 degrees equals 1/2 the difference between the measurement of

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the far arc – that’s our unknown…

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…and the near arc measurement of 60 degrees here.

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Multiply both sides of the equation by 2 to cancel out the 1/2 on the right side.

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45 times 2 equals 90, so you get 90 degrees on the left side...

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equals arc A-B-C minus 60 degrees.

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Add 60 degrees to both sides of the equation and we have the solution:

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Arc A-B-C should be 150 degrees… if the experiment goes as intended.

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Now that all of the theoretical measurements are in order, Roger’s ready to begin the test.

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Perhaps he should have factored in a few more variables… like… actual geese, for example.

01:53

This has turned into a real honk-fest.

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