WEBVTT
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𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐵𝐶 is equal to three centimeters and 𝐴𝐵 is equal to four centimeters.
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Find the length of line segment 𝐴𝐶 and the measures of angle 𝐴 and angle 𝐶 to the nearest degree.
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We will begin by sketching the right triangle 𝐴𝐵𝐶.
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We are told that side 𝐵𝐶 is three centimeters long and side 𝐴𝐵 is four centimeters long.
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The first part of our question is to find the length of the line segment 𝐴𝐶.
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This is the hypotenuse of the triangle, since it is opposite the right angle.
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One way of calculating the length of the hypotenuse when given the length of the other two sides of a right triangle is using the Pythagorean theorem.
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This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the two shorter sides.
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Substituting in the values from this question, we have 𝐴𝐶 squared is equal to three squared plus four squared.
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Three squared is equal to nine, and four squared, 16.
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We can then square root both sides of our equation.
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And since 𝐴𝐶 must be positive, we have 𝐴𝐶 is equal to the square root of nine plus 16.
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This simplifies to the square root of 25, which is equal to five.
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The length of the line segment 𝐴𝐶 is five centimeters.
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It is worth noting that this triangle is an example of a Pythagorean triple.
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And as a result, we may simply have recalled that any triangle with side lengths three centimeters, four centimeters, and five centimeters will be a right triangle.
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The next part of this question is to find the measures of angles 𝐴 and 𝐶.
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We will do this using our knowledge of right angle trigonometry and the sine, cosine, and tangent ratios.
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One way of recalling these ratios is using the acronym SOH CAH TOA, where sin 𝜃 is equal to the opposite over the hypotenuse, cos 𝜃 is equal to the adjacent over the hypotenuse, and tan 𝜃 is equal to the opposite over the adjacent.
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We will now use one of these ratios to help calculate the measure of angle 𝐴.
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The side of our triangle that is opposite this angle is 𝐵𝐶, and the side that is adjacent to the angle is the side 𝐴𝐵.
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We have already labeled the longest side 𝐴𝐶 as the hypotenuse.
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As we know all three lengths, we can use any one of the three ratios.
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In this question, we will choose to use the tangent ratio.
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The tangent of any angle is equal to the opposite over the adjacent.
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So, in this question, tan 𝐴 is equal to three over four, or three-quarters.
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To solve for 𝐴, we take the inverse tangent of both sides such that 𝐴 is equal to the inverse tan of three-quarters.
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Ensuring that our calculator is in degree mode, we can type this in, giving us an answer of 36.8698 and so on.
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We are asked to give our answer to the nearest degree, so this is equal to 37 degrees.
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The measure of angle 𝐴 is 37 degrees.
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In order to calculate the measure of angle 𝐶, we could once again use one of our trigonometric ratios.
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However, it is important to note that side length 𝐴𝐵 is now the opposite, as it is opposite angle 𝐶.
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In a similar way, side length 𝐵𝐶 is now the adjacent side.
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Once again, we could use any one of the three ratios.
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Using the tangent ratio, we have tan 𝐶 is equal to four over three.
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Taking the inverse tangent of both sides, 𝐶 is equal to the inverse tan of four-thirds.
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And rounded to one decimal place, this is equal to 53 degrees.
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The measure of angle 𝐶 is equal to 53 degrees.
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At this stage, it is worth checking that our three angles sum to 180 degrees, as this is true of any three angles in a triangle.
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The three answers to this question are the length of line segment 𝐴𝐶 equals five centimeters, the measure of angle 𝐴 is 37 degrees, and the measure of angle 𝐶 is 53 degrees.