PQRS is a Quadrilateral Whose Diagonals are Perpendicular to Each Other GMAT Problem Solving

Question: PQRS is a quadrilateral whose diagonals are perpendicular to each other. If PQ = 16cm, QR = 12cm, and RS = 20cm, what is the value of PS?

1. \(8\sqrt2\)
2. \(12\sqrt2\)
3. \(16\sqrt2\)
4. \(20\sqrt2\)
5. \(24\sqrt2\)

Answer:

Approach Solution (1):

From the above figure:

We have 4 right angled triangles PTQ, QTR, RTS, and STP

Let QT = a, PT = b, RT = c, and ST = d

\(\rightarrow16^2=a^2+b^2.....(1)\)

\(\rightarrow12^2=a^2+c^2.....(2)\)

\(\rightarrow20^2=c^2+d^2.....(3)\)

\(\rightarrow{x^2}=b^2+d^2.....(4)\)

\((1)+(3) \)

\(\rightarrow16^2+12^2=a^2+b^2+c^2+d^2\)

\(and \)

\((2)+(4)\)

\(\rightarrow12^2+x^2=a^2+b^2+c^2+d^2\)

\(\rightarrow16^2+20^2=a^2+b^2+c^2+d^2\)

\(256+400=144+x^2\)

\(x^2=656-144=512\)

\(x=16\sqrt2\)

Correct option: C

Approach Solution (2):

Since PQRS quadrilateral has two diagonals perpendicular to each other, four right angled triangles are formed right angled at point of intersection O.

Now

\(PQ^2=PO^2+OQ^2\)

\(16^2=PO^2+OQ^2......Eqn.1\)

\(QR^2=QO^2 +OR^2\)

\(12^2=QO^2+OR^2......Eqn. 2\)

\(RS^2=RO^2+OS^2 \)

\(20^2=RO^2+OS^2.......Eqn. 3\)

\(SP^2=SO^2+OP^2?\)

Adding Eqn. (1) + (2) + (3)

\(16^2+12^2+20^2=PO^2+OQ^2+QO^2+OR^2+RO^2+OS^2\)

\(256+144+400=PO^2+OS^2+2(OQ^2+OR^2)\)

\(256+144+400=PO^2+OS^2+2∗144\)

\(256+144+400−2∗144=PO^2+OS^2\)

\(PO^2+OS^2=256+400−144=512\)

\(PO^2+OS^2=2^9=(2^4∗2^{\frac{1}{2}})^2\)

\(PO^2+OS^2=16\sqrt2\)

Correct option: C

“PQRS is a quadrilateral whose diagonals are perpendicular to each other. If PQ = 16cm, QR = 12cm, and RS = 20cm, what is the value of PS?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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