# Question Video: Calculating an Unknown Length Using the Law of Cosines Mathematics • 11th Grade

A man ran 24 km down a straight road. He then turned 139° and ran a further 9 km along another straight road. Find to the nearest kilometer the shortest distance between his start and end points.

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### Video Transcript

A man ran 24 kilometers down a straight road. He then turned 139 degrees and ran a further nine kilometers along another straight road. Find to the nearest kilometer the shortest distance between his start and end points.

We are told that the man originally runs 24 kilometers along a straight road. He then turns 139 degrees and runs a further nine kilometers. We are asked to calculate the shortest distance between his start and end points.

We know that angles on a straight line sum to 180 degrees. And 180 minus 139 is equal to 41. This means that the angle inside our triangle is 41 degrees. We can now calculate the shortest distance labeled 𝑥 using the cosine rule. This states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by cos 𝐴. The side we’re trying to calculate is opposite the given angle. Substituting in our values, we have 𝑥 squared is equal to 24 squared plus nine squared minus two multiplied by 24 multiplied by nine multiplied by cos of 41 degrees.

Typing the right-hand side into the calculator gives us 330.9654 and so on. We then need to square root both sides of this equation. This gives us 𝑥 is equal to 18.1924 and so on. We are asked to give our answer to the nearest kilometer. Therefore, the one in the tenth column is the deciding number. As this is less than five, we round down. The shortest distance between the start and end points to the nearest kilometer is 18 kilometers.