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# Geometry

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In triangle PQR, let M be the midpoint of QR, let N be the midpoint of PR, and let O be the intersection of QN and RM, as shown. If QN perp PR, QN = 12, and PR = 14, then find the area of triangle PQR.

Feb 10, 2023

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Since QN is perpendicular to PR, we know that triangle QNO is a right triangle. Using the Pythagorean theorem, we can find the length of QO:

\(QO = sqrt(QN^2 + PR^2) = sqrt(12^2 + 14^2) = sqrt(288) = 12√2\)

Now, we can find the length of RO by adding the length of QO and NO:

\(RO = QO + NO = 12√2 + 7 = 19√2\)

Since triangle PQR is isosceles with midpoints M and N, we know that PM = PR/2 = 7 and PQ = PR = 14.

Let's call the height of triangle PQR "h". Then, using the area formula for triangle PQR, we get:

\(Area = (1/2)bh = (1/2)(14)(h)\)

Using the Pythagorean theorem, we can find h in terms of RO:

\(h^2 = PQ^2 - RO^2 = 14^2 - (19√2)^2\)

So,

\(h = sqrt(14^2 - (19√2)^2) = sqrt(196 - 368) = sqrt(-172)\)

Since the square of a real number is always non-negative, we know that this value of h is not possible. Therefore, triangle PQR does not exist.

Feb 10, 2023