The distances from a point to the corners of a rectangle are $6$, $7$, $9$, and (integer) $d$. Find $d$.

Question

Christina is standing in a rectangular garden. Her distances from the corners of the garden are $6$ meters, $7$ meters, $9$ meters, and $d$ meters, where $d$ is an integer. How to find $d$?

Can someone lend me your hand on it?

Answer 1

HINT

We have 5 unknown to determine, then we can guess at first $6$ opposite to $d$

with the following conditions

• $d^2-y^2=49-x^2$
• $49-w^2=36-z^2$
• $36-x^2=81-y^2$
• $81-z^2=d^2-w^2$

and then trying others configurations.

Answer 2

Let $P$ be a point inside a rectangle $XYZW$ such that $\{PX,PY,PZ\}=\{6,7,9\}$ and $PW=d$ is an integer. Suppose that the projections of $P$ onto $XY$, $YZ$, $ZW$, and $WX$ are $A$, $B$, $C$, and $D$, respectively. Write $$x:=XA=WC\,,\,\,y:=YA=ZC\,,\,\,z:=YB=XD\,,\text{ and }w:=ZB=WD\,.$$ Therefore, $$PX^2=x^2+z^2\,,\,\,PY^2=y^2+z^2\,,\text{ and }PZ^2=y^2+w^2\,.$$ This gives $$d^2=PW^2=x^2+w^2=(x^2+z^2)+(y^2+w^2)-(y^2+z^2)=PX^2+PZ^2-PY^2\,.$$ The only possible choices for $(PX,PY,PZ)$ such that $d$ is an integer are $$(PX,PY,PZ)=(6,9,7)\text{ or }(PX,PY,PZ)=(7,9,6)\,,$$ for which $d=2$. Below is an example of a possible configuration.

Answer 3

We have a relation using Stewart's theorem for an arbitrary parallelogram as follows "if ABCD is a parallelogram and P be any point in plane then $2AM^2 +2CM^2 - 2BM^2 + 2DM^2 $ = $ (AC^2 - BD^2)$ .

For a rectangle AC =BD then by proceeding, we can get desired un known.