WEBVTT
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๐ด๐ต๐ถ is a right-angled triangle at ๐ต.
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The point ๐ท lies on the vector ๐ต๐ถ, where ๐ถ๐ท is equal to 17 centimetres, the measure of the angle ๐ด๐ท๐ถ is 46 degrees, and the measure of the angle ๐ถ๐ด๐ท is 24 degrees.
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Find the length of ๐ด๐ต, giving your answer to the nearest centimetre.
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The clue here is that the point ๐ท lies on the vector ๐ต๐ถ.
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Thereโs actually no way the point can lie between ๐ต and ๐ถ and still be 46 degrees.
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Letโs sketch this out and see what it looks like.
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Notice that we know two of the angles in the triangle ๐ด๐ถ๐ท and the length of one of its sides.
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This means we can use the law of sines to calculate the length of the side ๐ด๐ถ: thatโs the side shared by both of the triangles.
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We know that we need to use the law of sines over the law of cosines since that law requires at least two known sides.
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Now, we can use either of these forms for the sine rule.
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Since weโre trying to find a side though, itโs sensible to use the first form in order to minimize the amount of rearranging we need to do.
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The second form for the law of sines is better when weโre trying to calculate the measure of one of the angles.
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Letโs begin by labelling the sides of this triangle.
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The side ๐ sits directly opposite the angle ๐ด, the side ๐ sits directly opposite angle ๐ถ, and the side ๐ sits directly opposite the angle ๐ท.
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The rule will change slightly to ๐ over sin ๐ด equals ๐ over sin ๐ถ equals ๐ over sin ๐ท to account for the names of the angles in our triangle.
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Since weโre using the side ๐ and trying to find the side ๐, weโll use these two parts of the equation.
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Substituting the relevant values into our equation gives us 17 over sin of 24 equals ๐ over sin of 46.
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Weโll then solve this equation by multiplying both sides by sin of 46.
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And that gives us that ๐ is equal to 17 over sin of 24 multiplied by sin of 46.
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Popping this into our calculator, we get a value of 30.065 and so on.
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We wonโt round this answer just yet in order to prevent any mistakes from rounding too early.
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Weโve calculated the length of the side ๐ด๐ถ.
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Thatโs the length of the hypotenuse of this right-angled triangle.
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We can use right angle trigonometry to find the length ๐ด๐ต, which Iโve labelled as ๐ฅ.
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Before we do though, we can calculate the measure of the acute angle at ๐ถ.
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Remember this angle is equal to the sum of the angles at ๐ด and ๐ท.
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The exterior angle in a triangle will always be equal to the sum of the two interior opposite angles.
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Thatโs 70 degrees.
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Now, letโs label the triangle ๐ด๐ต๐ถ using the conventions for right angle trigonometry.
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The length ๐ด๐ถ is the hypotenuse of the triangle.
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And weโre trying to find the length ๐ด๐ต, which is the opposite side.
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Itโs the side opposite the angle we just calculated.
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In this case then, we can use the sine ratio to help us find the length that weโve labelled ๐ฅ.
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Substituting what we know from our triangle into this formula gives us sin of 70 equals ๐ฅ over 30.065.
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Then, we can solve this equation by multiplying both sides by our nonrounded number 30.065.
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Thatโs ๐ฅ is equal to sin of 70 multiplied by 30.06 and so on.
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And when we type it into our calculator, we get a value of 28.252.
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Correct to the nearest centimetre, the length of ๐ด๐ต is 28 centimetres.