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 = 11 yards, b = 9 yards, c = 7 yards

Accepted Answer

Hey, everyone, here we are asked to determine the area of the triangle using Heron's formula and report the area rounded to the nearest square unit. Here we are given the sign lengths of our triangle. We have side length P which is 21 ft. We are then given side length Q which is 16 ft. And lastly, we have a side length R which is 6 ft. We have four answer choice options, answer choice, A 30 square feet, answer B 31 square feet, answer C 52 square feet and answer D 53 square feet. So here we can notice we are given all of our side links for our given triangle. So to determine the area, we need to recall Huron's formula. And so Huron's formula states that the area is equivalent to the square root of S applied by S minus a multiplied by S minus B multiplied by S minus C. And so now taking a closer look at this formula, we can notice variables A B and C. Well, these refer to these side lengths of our triangle, but we notice that we have the variables PQ and R for our given side lengths. So we can have side length A B equivalent to side length P. Then we have side length B will be equivalent to side length Q. And lastly, we'll, we will have side length CB, side length R. And lastly, we can notice that we have another variable which is variable s this variable represents the semi perimeter of the triangle. And it has its own formula which we, we can recall the semi perimeter is equivalent to A plus B plus C all divided by two. And now just using our variables given, we know that the semi perimeter is equivalent to P plus Q plus R all divided by two. And so now we can go ahead and calculate the semi perimeter by just substituting in our given values. So we have 21 plus plus six all divided by two. And now just simplifying, we find that S is equivalent to 21.5 ft. So now, since we have calculated S and we are given the rest of our values, we can go ahead and just substitute these values into our formula to solve for the area. So we have that the area is equal to the square root of 21.5 multiplied by 21.5 minus 21 multiplied by 21.5 minus 16, multiplied by 21.5 minus six. And now continuing to solve for our area, we just want to first simplify the values underneath our square root. So performing all of the subtraction operations, we are left with the square root of 21.5 multiplied by 0.5 multiplied by 5.5 multiplied by 15.5. And now multiplying all of our terms together here we have the square root of the value 916.4375. And now with this value, we cannot simplify our expression any further. So we just want to find the decimal approximation of this value which is 30.27. And again, we are asked to round our answer to the nearest square unit. So just rounding this value, we find that our area comes out to just square feet. And so with this value, we are left with answer choice. A where again, the area of our given triangle is 30 square feet. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.a = 11 yards, b = 9 yards, c = 7 yards

Key Concepts

Heron's Formula

Semi-perimeter

Triangle Inequality Theorem

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