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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.

Mar 5, 2023

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Since QN is perpendicular to PR, triangle QNR is a right triangle. Using the Pythagorean theorem, we can find the length of QR:

QR^2 = QN^2 + NR^2
QR^2 = 12^2 + (PR/2)^2
QR^2 = 144 + 49
QR^2 = 193
QR = sqrt(193)

Now we can use the fact that M and N are midpoints to find the lengths of PM and RN:

PM = QR/2 = sqrt(193)/2
RN = PR/2 = 7

Since O is the intersection of QN and RM, we can use similar triangles QON and QRP to find the length of ON:

ON/QN = PR/QR
ON/12 = 14/sqrt(193)
ON = 168/sqrt(193)

Now we can use the fact that O is the midpoint of RM to find the length of RM:

RM = 2*ON = 336/sqrt(193)

Finally, we can use the formula for the area of a triangle in terms of its base and height:

A = (1/2)bh

where b is the base and h is the height.

The height of triangle PQR is RN = 7. To find the base, we can use the fact that O is on RM, so the base is PR - PM - MR:

b = PR - PM - MR
b = 14 - sqrt(193)/2 - 336/sqrt(193)

Substituting these values into the formula for the area, we have:

A = (1/2)bh
A = (1/2)(14 - sqrt(193)/2 - 336/sqrt(193))(7)
A = (49/2) - (7/2)sqrt(193) - 168

Therefore, the area of triangle PQR is (49/2) - (7/2)sqrt(193) - 168.

Mar 6, 2023