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CAT 2019 Slot 2 QA Question & Solution

GeometryEasy

Question

Let ABC be a right-angled triangle with hypotenuse BC of length 20 cm. If AP is perpendicular on BC, then the maximum possible length of AP, in cm, is

Options

10
5
$8\surd2$
$6\surd2$

Solution

Question Figure

Let p be the length of AP.

It is given that $\angle\ BAC\ =90$ and $\angle\ APC\ =90$

Let $\angle\ ABC\ =\theta$, then $\angle\ BAP\ =90-\theta$ and $\angle\ BCA\ =90-\theta$

So $\angle\ PAC\ =\theta$

Triangles BPA and APC are similar

$p^2=x\left(20-x\right)$

We have to maximize the value of p, which will be maximum when $x=20-x$

$x=10$