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Given that π΄πΆ equals eight centimeters and the radius equals eight centimeters, find the area of triangle π΄π΅πΆ rounded to the nearest integer.
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And then we have a diagram which shows a circle with a triangle inscribed within it.
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We notice that line π΄π΅ passes through the center of the circle, and so line π΄π΅ is the diameter.
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So before we work out the area of the triangle weβve been given, letβs identify what extra information we can calculate given the information weβve been given.
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We are told that π΄πΆ equals eight centimeters and that the radius is equal to eight centimeters.
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This means line segments π΅π and ππ΄ are both eight centimeters.
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Now, in fact, since point π is the center of the circle and πΆ lies on the circumference, we see that ππΆ also forms the radius of our circle.
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So ππΆ is also eight centimeters.
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And this is really useful since we know that all angles in an equilateral triangle, that is, a triangle where all three sides are of equal length, are 60 degrees.
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But this doesnβt necessarily help us find the area of triangle π΄π΅πΆ.
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Remember, the formula that we can use to find the area of a triangle is a half times base times height.
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So if we could work out the length of the base and the height of this triangle π΄π΅πΆ, weβd be able to find its area.
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Now, in fact, we can quite quickly work out which sides of our triangle represent the base and the height.
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The base and the height must be perpendicular to one another.
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So weβre interested in sides π΄πΆ and π΅πΆ.
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But how do we know that?
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Well, we know that the angle subtended by the diameter is 90 degrees.
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Our angle π΅πΆπ΄ is indeed subtended from the diameter, so π΅πΆπ΄ is 90 degrees.
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So we can define π΄πΆ to be the base of our triangle and that is eight centimeters in length.
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But then this means that π΅πΆ is the height of our triangle.
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So what is the length of π΅πΆ?
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Well, now that we know we have a right triangle and we know a couple of the angles given in this triangle, we can work out the length of π΅πΆ using right-triangle trigonometry.
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Here is our triangle π΄π΅πΆ with the right angle at πΆ.
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We calculated that the measure of angle π΄ is 60 degrees.
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And of course, weβre trying to find the length of π΅πΆ, so letβs define that to be equal to π₯ centimeters.
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This side of the triangle lies directly opposite the included angle of 60 degrees, whilst the side π΄πΆ is adjacent to the angle.
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And so we need to identify the trigonometric ratio that links the opposite side with the adjacent, that is, the tangent ratio.
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tan π is opposite over adjacent.
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In this case then, tan of 60 is π₯ divided by eight.
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But of course, tan of 60 is one of our exact value ratios.
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Itβs the square root of three, so we get root three equals π₯ divided by eight.
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And if we multiply through by eight, we find that π₯ is equal to eight root three, and thatβs great.
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We now know the length of π΅πΆ, which we said was the height of our triangle.
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Itβs eight root three centimeters.
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So the area of our triangle is one-half times eight times eight root three.
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In exact form, that gives us 32 root three square centimeters.
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But If we enter this into our calculator, correct to the nearest integer, thatβs 55 square centimeters.
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And so the area of triangle π΄π΅πΆ correct to the nearest integer is 55 square centimeters.
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Now, itβs worth noting that we didnβt actually need to use right-triangle trigonometry.
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Instead, if weβd recognize that π΄π΅ is the diameter and therefore 16 centimeters in length and we already knew that triangle π΄πΆπ΅ was a right triangle at πΆ, we could have used the Pythagorean theorem to calculate the length of π΅πΆ.
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In that case, we once again would have got eight root three, giving us a final area of 55 square centimeters.
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Either method is perfectly valid.