Proper Alignment

Geometry Level 3

$WXYZ$ is a square of side length 30. $V$ is a point on $XY$ and $P$ is a point inside square with $PV$ perpendicular to $XY$ , also $PW = PZ = PV-5$ . What is $PV$ ?

The answer is 22.

2 solutions

Ahmad Saad
Aug 14, 2016

Applying Pythagoras theorem on $PQW$ , we obtain

$( S - 5)^2 = (30-S)^2 + 15^2$

Expanding and simplifying, we get $50 S = 1100$ , or that $S = 22$ .

The diagram in the question should be more perfectly scaled.

Aniruddha Bagchi - 4 years, 10 months ago

how did you know that VY was 15 ?

Gamael Marcel - 4 years, 10 months ago

PW = PZ ---> Tr. PWZ is isosceles ---> its altitude PQ bisects its base WZ

PV perpendicular to XY ---> points Q,P and V are collinear ---> V bisect XY

VY = XY/2 = 15

Ahmad Saad - 4 years, 10 months ago

Nice solution. I have a question. How can you draw this shape?

Utku Demircil - 4 years, 9 months ago

the shape is drawn by AutoCad software. You can get it from websites.

Ahmad Saad - 4 years, 9 months ago

Why do you joined VZ .

D H - 4 years, 9 months ago

D H
Aug 24, 2016

Let PW = PZ = x

Draw PM perpendicular ZW .

In an isosceles triangle perpendicular from the vertex bisects the base .

WM = 15

√ (x^2 - 225 ) + (x + 5) = 30

=> √ (x^2 - 225 ) = 30 - 5 - x

=> (x^2 - 225 ) = ( 25 - x ) ^2 {squaruing both sides )

=> (x^2 - 225 ) = 625 - 2 * 25 * x + x^ 2

=> 50x = 850

=> x = 17

=> PV = 17 + 5 = 22 Q.V.E.D by Vishwash .....................

Proper Alignment

The answer is 22.

2 solutions

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