*without*using Heron's Formula. To wit:

To find the area of triangle PQR we need a base and a height. If we consider side PR the base of the triangle, then QS, which is at a right angle to PR, is the height. We know that both triangle PQS and triangle SQR are right triangles, but to find the height QS we’ll need to know the length of either PS or SR. Unfortunately we don’t know either one, so we’ll have to name variables for several legs of the triangle.

In this case we’ll let x represent the length of PS and y represent the height QS. Note that if PS has length x, then SR has length 21 – x, so we do not need to (and should not) name a variable for length SR. Updating our diagram yields the following:

Apply the Pythagorean Theorem to each right triangle.

From triangle PQS: x^{2}+ y^{2}= 100

From triangle SQR: (21 – x)^{2}+ y^{2}= 289

We solve each in terms of y^{2}:

y^{2}= 100 – x^{2}

y^{2}= 289 – (21 – x)^{2}

Then set the two expressions equal to y^{2}equal to each other:

289 – (21 – x)^{2}= 100 – x^{2}

Now simplify:

189 – (21 – x)^{2}= -x^{2}

189 = (21 – x)^{2}– x^{2}

189 = (441 – 42x + x^{2}) – x^{2}

189 = 441 – 42x

42x = 252

x = 6

Now that the value of x is known, solve for y:

100 = 36 + y^{2}

64 = y^{2}

8 = y

(Or just recognize that PQS is a 6 – 8 – 10 right triangle.)

Finally, we can solve for the area of triangle PQR:

(1/2)bh = (1/2)(21)(8) = 84

The correct answer is B.

{Incidentally, this problem is named for Heron’s Formula, which is an alternative to the (1/2)bh triangle area formula. For a triangle with side lengths a, b, and c, the semiperimeter is defined as s = (a + b + c)/2. The area of the triangle equals √(s(s-a)(s-b)(s-c)). Note that this doesn’t require a right triangle, nor does it require knowledge of any heights measured perpendicular to any base. You will NOT need to know this for the GRE, as the original solution above attests.}

I was glad to see the above explanation, because I was honestly stumped as to how to approach the problem. I had thought that Heron's Formula would be the only way to solve it.

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