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๐ด๐ต๐ถ is a triangle where ๐ด๐ต equals 50 centimeters, ๐ต๐ถ equals 30 centimeters, and ๐ด๐ถ equals 42 centimeters.
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Find the area of the triangle, giving the answer to the nearest square centimeter.
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To help us understand whatโs happening with this question, Iโve just drawn a little sketch of a triangle.
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Doesnโt necessary represent the scale or the right proportion of our triangle.
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But what I wanna do is just give us an idea of where the sides are going to be.
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So we know that ๐ด๐ต is equal to 50 centimeters.
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๐ต๐ถ is 30 centimeters.
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And ๐ด๐ถ is 42 centimeters.
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Okay, so now what we want to actually do is we want to find the area of this triangle.
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However, we donโt know which is the base, which is the perpendicular height.
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We donโt know anything like this.
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So how are we actually going to do it?
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Well, to enable us to actually find the area of this triangle, what weโre actually going to use is Heronโs formula.
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And Heronโs formula tells us that ๐ด, the area of the triangle, is equal to the square root of ๐ multiplied by ๐ minus ๐ multiplied by ๐ minus ๐ multiplied by ๐ minus ๐, where ๐ is equal to our semiperimeter.
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So thatโs the semiperimeter of our triangle.
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And we have a formula for this.
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And thatโs ๐ is equal to ๐ plus ๐ plus ๐, because that would be our perimeter of our triangle, and then all divided by two because itโs a semiperimeter.
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Okay, great!
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Weโve got the formula we need.
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So now letโs go and find the area of our triangle.
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So first of all, weโre actually going to find ๐ .
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And to do that, weโre gonna substitute in our values for ๐, ๐, and ๐.
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So then when we substitute in our values, we get 30 plus 42 plus 50 all divided by two.
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Great!
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So now we can calculate ๐ .
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So ๐ is equal to 122 over two, which is equal to 61.
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Okay, weโve found our semiperimeter.
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Now letโs go on and find our area.
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And to find our area, weโre actually gonna substitute our values into Heronโs formula.
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So we have ๐ด, our area, is equal to the square root of 61, because that was our ๐ , that was our semiperimeter.
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And then this is multiplied by 61 minus 30, cause 30 is our ๐, multiplied by 61 minus 42 multiplied by 61 minus 50, which gives us the square root of 61 multiplied by 31 multiplied by 19 multiplied by 11.
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And then we get a final answer of 628.664.
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Okay, not quite finished there.
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Thereโs one more thing to do.
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The question asks for the answer to the nearest square centimeter.
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So therefore, we can say that the area of the triangle ๐ด๐ต๐ถ, where ๐ด๐ต is equal to 50 centimeters, ๐ต๐ถ is equal to 30 centimeters, and ๐ด๐ถ is equal to 42 centimeters, is equal to 629 centimeters squared to the nearest square centimeter.