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GATE 2022 CE (CE1) – Question 5

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

In the following diagram, the point R is the center of the circle. The lines PQ and ZV are tangential to the circle. The relation among the areas of the squares, PXWR, RUVZ and SPQT is

Circle with centre R, tangent PQ, and squares SPQT, PXWR and RUVZ erected on the three sides of right triangle PQR. See the attached source image.
  1. Area of SPQT = Area of RUVZ = Area of PXWR
  2. Area of SPQT = Area of PXWR − Area of RUVZ
  3. Area of PXWR = Area of SPQT − Area of RUVZ
  4. Area of PXWR = Area of RUVZ − Area of SPQT

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Show answer and explanation

Correct answer: (B) Area of SPQT = Area of PXWR − Area of RUVZ

Explanation

The radius RQ is perpendicular to tangent PQ, so triangle PQR is right-angled at Q. Pythagoras gives $PR^2=PQ^2+RQ^2$.

The three squares have areas $PR^2$, $PQ^2$ and $RQ^2$, respectively. Rearranging, **area SPQT = area PXWR − area RUVZ**, option **B**.