In Exercises 25–30, use Heron's formula to find the area of each ...

Question

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.

a = 14 meters, b = 12 meters, c = 4 meters

Accepted Answer

Hello, everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states for the given triangle determine the area of the triangle using Huron's formula and report the area rounded to the nearest square unit. Now, they tell us that the sides of the triangle are P equals 24 degree centimeters. Q is equal to 15 centimeters and R is equal to 11 centimeters. Now, answer choice A says that the answer will be 58 square centimeters. B has 95 square centimeters. C says 60 square centimeters and D says 59 square centimeters. Now, the first thing we have to think about is what is hero's formula. So let's write it every recall. Huron's formula states that the area will be equal to the square root of S multiplied by open parentheses. S minus A closed parentheses, multiplied by open parentheses. S minus B closed parentheses multiplied by open parentheses. S minus C close parentheses and all that will be within the square roots. Now, this is where A B and C our sides and S is the semi perimeter. Now, they told us the values of all the sides. So all we have to do is calculate the semi perimeter, the S. And luckily, we have a formula for that, we know that the semi perimeter is equal to a fraction when the numerator, we have A plus B plus C and that will be divided by two. So the semi perimeter is just half of the perimeter. We know that perimeter is all the sides added up. Now, the semi perimeter is just that divided by two. So we'll have that S is equal to, in the numerator, we have 24 plus 15 plus 11. Those are the values of the sides that were given to us. And we'll have all of that divided by two, which will be equal to 50 divided by two, which is equal to 25. And in this case, that will be 25 centimeters. Now, we can plug that into Huron's formula. So we'll have that area A is equal to the square root of S is 25. So we have 25 multiplied by open parentheses. 25 minus 24. This is S minus the first side given closed parentheses and then multiplied by open parentheses, 25 minus 15, which is the other side, closed parentheses, multiplied by open parentheses 25 minus 11, which is the last side and we'll close that parentheses and all of that will be within the square root. So now, all we have to do is simplify and we'll have that area A is equal to the square root of 25 multiplied by one, multiplied by 10, multiplied by 14. And all we did there was just simplify all the parentheses that we had. Now, if we distribute and multiply everything together, we'll have A is equal to the square roots of 3500 which means that A is around 59.16. Once we plug that into our calculator, and if we round that out to the nearest whole square unit, we have 59 square centimeters or centimeters squared. And that will be our answer for this question which matches with answer choice D. So you can see how a simple formula can help us find the area of our triangle. So I hope that was helpful. And until next time.

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit. a = 14 meters, b = 12 meters, c = 4 meters

Key Concepts

Heron's Formula

Semi-perimeter

Triangle Inequality Theorem

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