PQRS is a square. M is the mid point of side PQ and N is a point on QR such that NR is two thirds of QR. If the area of triangle MQN is 48 sq cm, what is the length (in cm) of diagonal PR?

Introduction / Context:
This question mixes coordinate style reasoning with properties of a square. By placing the square on a coordinate grid, we can express the coordinates of key points, compute the area of a triangle in terms of the side length, then use the given area to find the side and finally the diagonal. It is a standard application of area formulas and basic analytic geometry ideas.

Given Data / Assumptions:

• PQRS is a square with side length a (unknown).
• M is the mid point of PQ.
• N is on QR such that NR is two thirds of QR.
• Area of triangle MQN is 48 sq cm.
• We must compute the length of diagonal PR.

Concept / Approach:
We can set up convenient coordinates for the vertices of the square, for example P at (0, 0), Q at (a, 0), R at (a, a) and S at (0, a). From there, the coordinates of M and N can be written using mid point and segment division facts. Using the determinant or coordinate area formula for a triangle, we get the area of triangle MQN in terms of a. Setting this equal to 48 allows solving for a. The diagonal PR of the square then has length a√2.

Step-by-Step Solution:
Step 1: Place the square so that P = (0, 0), Q = (a, 0), R = (a, a) and S = (0, a). Step 2: M is the midpoint of PQ, so M = ((0 + a) / 2, (0 + 0) / 2) = (a / 2, 0). Step 3: NR is two thirds of QR, so QN is one third of QR. Since QR goes from (a, 0) to (a, a), point N is at (a, a / 3). Step 4: Coordinates are M(a / 2, 0), Q(a, 0) and N(a, a / 3). Step 5: Area of triangle MQN using base along MQ: base MQ = a − a / 2 = a / 2 and height is a / 3, so Area = (1 / 2) * (a / 2) * (a / 3) = a^2 / 12. Step 6: Given area is 48 sq cm, so a^2 / 12 = 48 which gives a^2 = 48 * 12 = 576 and a = 24 cm. Step 7: Diagonal PR of a square of side a is a√2, so PR = 24√2 cm.

Verification / Alternative check:
We can verify the area calculation by using the coordinate determinant formula for triangle area. Using points M(a / 2, 0), Q(a, 0) and N(a, a / 3), the absolute value of the determinant divided by 2 again yields a^2 / 12. Substituting a = 24 gives area = 24^2 / 12 = 576 / 12 = 48 sq cm, which matches the question, and then PR = 24√2 cm follows directly. This confirms that the reasoning and arithmetic are correct.

Why Other Options Are Wrong:
A diagonal of 12√2 cm or 12 cm corresponds to a much smaller square with side 12 cm or less, which would give a triangle area significantly less than 48 sq cm. A diagonal of 24 cm would imply side length less than 24 cm, again incompatible with the computed triangle area. The extra option 18√2 is intermediate but does not match the required side length from the area equation. Only 24√2 cm is consistent with the given area and square geometry.

Common Pitfalls:
It is easy to misinterpret the condition NR equals two thirds of QR and accidentally place the point N at the wrong location. Another common error is miscomputing the triangle area or forgetting that the base and height chosen must be perpendicular. Drawing a clear diagram, assigning coordinates carefully and then using standard formulas step by step helps avoid mistakes in such coordinate geometry style problems.

Final Answer:
The length of diagonal PR is 24√2 cm.