How much does it occupy?

Geometry Level 4

Let, $d(P,AB)$ denote the distance perpendicular between the point $P$ and the line $AB$ .

Let $OABC$ be a rectangle with $O$ as the origin and $A \equiv (3,0)$ , $B \equiv (3,2)$ , $C \equiv (0,2)$ . And let $P$ be point in the rectangle that is subjected to

$d(P,OA) < \text{min} \left[d(P,AB),d(P,BC),d(P,OC)\right]$

Find the area of the region of points satisfying the point $P$ .

The answer is 2.

1 solution

Surya Prakash
Aug 1, 2015

Let $D, E, F, G$ be the foot of perpendiculars from $P$ onto $OA, AB, BC, OC$ respectively.

Since, $PD < min \left[PE,PF,PG \right]$ . So, $P$ lies in the region of intersection of the regions $PD<PE$ , $PD < PF$ and $PD<PG$ . So, the region obtained is a trapezium with the parallel sides equal to $1$ and $3$ units respectively and with the height equal to $1$ unit.

So, Area required $= \dfrac{1}{2} \times 1 \times (1+3) =\boxed{2}$

Note : Here, $PD<PE$ is the region below the line $PD = PE$ .

Moderator note:

Good approach.

Just be careful with the labelling of the vertices in the images.

Note that your image does not have the vertices labelled correctly. It makes it harder to follow through what you're trying to say.

Calvin Lin Staff - 5 years, 10 months ago

thanks for the suggestion

Surya Prakash - 5 years, 10 months ago

How much does it occupy?

The answer is 2.

1 solution

Moderator note:

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