UPSC CDS 2 2021 Math Paper

Concept:It uses the converse of the Pythagorean theorem and the midpoint property to relate the given expression to the square of the hypotenuse.Explanation:Given: $\triangle ABC$ with $AB = 1.6$ cm, $BC = 6.3$ cm, $CA = 6.5$ cm.Check: $AB^2 + BC^2 = (1.6)^2 + (6.3)^2 = 2.56 + 39.69 = 42.25$. Also $CA^2 = (6.5)^2 = 42.25$.Since $AB^2 + BC^2 = CA^2$, $\triangle ABC$ is right-angled at $B$ (by Pythagoras theorem).Given: $AP = PB$ means $P$ is the midpoint of $AB$. $BQ = QC$ means $Q$ is the midpoint of $BC$.Then $BQ = \frac{BC}{2}$, so $4BQ^2 = 4 \times \left(\frac{BC}{2}\right)^2 = BC^2$.Thus, $AB^2 + 4BQ^2 = AB^2 + BC^2 = CA^2 = (6.5)^2 = 42.25$ cm$^2$.Answer:$42.25$ cm$^2$ (Option B).