UPSKILL MATH PLUS

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### Exercise condition:

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1. The perpendicular $$PS$$ on the base $$QR$$ of a $$\Delta PQR$$ intersects $$QR$$ at $$S$$, such that $$QS$$ $$=$$ $$3$$ $$SR$$. Prove that $$2$$ $$PQ^2 = 2PR^2 + QR^2$$.

2. In the adjacent figure, $$ABC$$ is a right-angled triangle with right angle at $$B$$ and points $$D$$, $$E$$ trisect $$BC$$. Prove that $$8AE^2$$ $$=$$ $$3AC^2 + 5AD^2$$.

3. $$P$$ and $$Q$$ are the mid-points of the sides $$CA$$ and $$CB$$ respectively of a $$\Delta ABC$$, right angled at $$C$$. Prove that $$4(AQ^2 + BP^2) = 5AB^2$$.