The question is from CAT Geometry - Trigonometry. This is about Trigonometric Identites. We need to find out the length from the given diagram. Trigonometry is an important topic for CAT Preparation for the CAT Exam. Trigonometric ideas can be completely opposite, as in, some questions test lot of common sense with direction, heights and distances, while others could test you on identities. CAT exam could test one on both these fronts. Make sure you master both in CAT - Trigonometry.

Question 6: In the below figure, the sheet of width W is folded along PQ such that R overlaps S Length of PQ can be written as :-

- \\frac{w}{sinα(1+cos2α)}\\)
- \\frac{w}{sin2αcosα}\\)
- \\frac{w}{cosα(1+sin2α)}\\)
- Any two of the above

Any two of the above

If you are quick at observing , this question can be solved just by looking at the options as ,

\\frac{w}{sin∝(1+cos2∝)}\\) = \\frac{w}{sin∝(1+cos^2∝ - sin^2∝}\\)

= \\frac{w}{2cos^2∝}\\) = \\frac{w}{sin2∝cos∝}\\) which is option (d)

∴ (d) Answer

Actually solving the question.

For R and S to overlap,

∆ PQR is congurent to ∆ PSQ

∴ ∠ QPR = ∝ ; ∠ SQP = ∠ PQR = 90 - ∝

=> ∠ RQT = 2∝ (180°-(180°-2∝))

We can draw the figure as :-

∴In ∆ QRT

QT = QR cos2∝

In ∆ PSQ,

SQ = PQ sin∝ = QR

∴ w = QT + SQ = PQ sin∝cos2∝ + PQ sin∝

= PQ sin∝ (1 + cos 2∝)

\\frac{w}{sin (1+cos2∝)}\\) = \\frac{w}{sin2∝cos∝}\\)(as shown earlier)

The question is **"Length of PQ can be written as "**

Choice D is the correct answer.

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