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Find the area of triangle π΄π΅πΆ, given that π΄π΅ is equal to π΄πΆ, π΅πΆ is equal to 20 centimeters, and cos of π΅ is equal to five over 13.
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The area of a triangle is found by multiplying its base by its perpendicular height and dividing by two.
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In this question, weβve only been given the length of one side of the triangle: π΅πΆ is 20 centimeters.
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In order to work out the area, we also need to know the perpendicular height of this triangle which Iβm going to refer to as π΄π·.
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The question tells us that two sides of the triangle π΄π΅ and π΄πΆ are the same length.
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And therefore, triangle π΄π΅πΆ is isosceles.
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This means when I draw in a perpendicular height from the shared vertex of the two sides of equal length to the opposite side, this divides the triangle up into two identical right-angled triangles.
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This means that the length of π΅πΆ 20 centimeters is divided exactly in half into two lengths of 10 centimeters.
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We still only know one length in each of these right-angled triangles.
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So letβs look at the other piece of information given in the question.
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Weβre told that cos or cosine of the angle π΅ is equal to five over 13.
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Remember the definition of the cosine ratio in a right-angled triangle is that cos of a particular angle π is equal to the length of the adjacent side divided by the length of the hypotenuse.
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Letβs label the three sides of the right-angled triangle π΄π΅π· in relation to angle π΅.
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The hypotenuse and longest side of our right-angled triangle is the side directly opposite the right angle.
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So thatβs side π΄π΅.
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The opposite is the side opposite the known angle.
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So thatβs side π΄π·.
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The adjacent is the final side.
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So thatβs the side between the known angle and the right angle, in this case π΅π·.
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Remember the cosine ratio tells us about the ratio between the adjacent and the hypotenuse.
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By substituting 10 for the length of the adjacent and π΄π΅ for the hypotenuse, we know then that cos of π΅ is equal to 10 over π΄π΅.
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This must also be equal to five over 13 as itβs stated in the question that cos of π΅ is equal to five over 13.
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This gives an equation that we can solve in order to find the length of π΄π΅.
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Now, ultimately, it isnβt π΄π΅ that we want to calculate, itβs π΄π·, the perpendicular height of the triangle.
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But we arenβt in a position to calculate π΄π· yet.
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However, if we can find π΄π΅ first, weβll then be able to calculate π΄π· afterwards.
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Cross multiplying will eliminate the two denominators in this equation, giving 10 multiplied by 13 is equal to five multiplied by π΄π΅.
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To find π΄π΅, we need to divide both sides of the equation by five, giving π΄π΅ is equal to 10 multiplied by 13 over five.
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A factor of five can be cancelled from both the numerator and denominator, giving two multiplied by 13 which is 26.
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We found then that the length of the side π΄π΅ is 26 centimeters.
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Remember ultimately, weβre looking to calculate the length of π΄π·.
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So letβs think about how we can do that now.
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We have a right-angled triangle, triangle π΄π΅π·, in which we know the length of two of the sides.
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This means that we can apply the Pythagorean theorem in order to calculate the length of the third side.
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The Pythagorean theorem tells us that in a right-angled triangle the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
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In our triangle, this means that π΄π· squared plus π΅π· squared is equal to π΄π΅ squared.
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Substituting the known length of π΅π· and π΄π΅ gives π΄π· squared plus 10 squared is equal to 26 squared.
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And we now have an equation that we can solve in order to find the length of π΄π·.
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10 squared is 100 and 26 squared is 676.
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So we have π΄π· squared plus 100 is equal to 676.
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Subtracting 100 from each side of the equation, we have that π΄π· squared is equal to 576.
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And then square rooting, we have that π΄π· is equal to 24.
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If youβre very familiar with Pythagorean triples β that is right-angled triangles in which all three sides are integers β you may have been able to spot this as 10, 24, 26 is an example of a Pythagorean triple.
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Whether you spotted it straightaway or whether you had to go through the working for the Pythagorean theorem, we now know that the perpendicular height of the triangle is 24 centimeters.
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And hence, weβre able to calculate the area.
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Remember the base of the full triangle π΄π΅πΆ is 20 centimeters.
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So we multiply 20 by 24 and divide by two.
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A factor of two can be cancelled from both the numerator and the denominator, giving 10 multiplied by 24 over one which is just equal to 240.
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The area of triangle π΄π΅πΆ is 240 centimeters squared.