Simple Looking Problem in Geometry
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The problem
Given the following figure
determine $x$.
My solution(s):
There are many possibilities to use similarities and the Pythagorean resulting more or less complicated systems of equations.
The simplest solution I could come up with was the following.
Because of the similarity of $AEF$ and $FCD$, we may say that $u=1/x$.
Based on the Pythagorean we have that
$$(1+x)^2+(1+1/x)^2=100.$$
This equation has two positive real solutions. One of them is:
$$x=frac12left(-1+sqrt101+sqrt98-2sqrt101right)approx 8.937.$$ The other one is $frac18.937.$
(http://www.wolframalpha.com/input/?i=solve+(1%2Bx)%5E2%2B(1%2B1%2Fx)%5E2%3D100)
The question
The source of the problem above is an old (Hungarian) high scholl level problem book. So, I suspect that there must be a simpler solution not requiring the roots of a fourth order equations. Please, either prove that there is no simpler solution or show one.
geometry euclidean-geometry
asked Sep 2 at 8:41
zoli
16.3k41643
add a comment |Â
up vote
0
down vote
favorite
The problem
Given the following figure
determine $x$.
My solution(s):
There are many possibilities to use similarities and the Pythagorean resulting more or less complicated systems of equations.
The simplest solution I could come up with was the following.
Because of the similarity of $AEF$ and $FCD$, we may say that $u=1/x$.
Based on the Pythagorean we have that
$$(1+x)^2+(1+1/x)^2=100.$$
This equation has two positive real solutions. One of them is:
$$x=frac12left(-1+sqrt101+sqrt98-2sqrt101right)approx 8.937.$$ The other one is $frac18.937.$
(http://www.wolframalpha.com/input/?i=solve+(1%2Bx)%5E2%2B(1%2B1%2Fx)%5E2%3D100)
The question
The source of the problem above is an old (Hungarian) high scholl level problem book. So, I suspect that there must be a simpler solution not requiring the roots of a fourth order equations. Please, either prove that there is no simpler solution or show one.
geometry euclidean-geometry
asked Sep 2 at 8:41
zoli
16.3k41643
Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The problem
Given the following figure
determine $x$.
My solution(s):
There are many possibilities to use similarities and the Pythagorean resulting more or less complicated systems of equations.
The simplest solution I could come up with was the following.
Because of the similarity of $AEF$ and $FCD$, we may say that $u=1/x$.
Based on the Pythagorean we have that
$$(1+x)^2+(1+1/x)^2=100.$$
This equation has two positive real solutions. One of them is:
$$x=frac12left(-1+sqrt101+sqrt98-2sqrt101right)approx 8.937.$$ The other one is $frac18.937.$
(http://www.wolframalpha.com/input/?i=solve+(1%2Bx)%5E2%2B(1%2B1%2Fx)%5E2%3D100)
The question
The source of the problem above is an old (Hungarian) high scholl level problem book. So, I suspect that there must be a simpler solution not requiring the roots of a fourth order equations. Please, either prove that there is no simpler solution or show one.
geometry euclidean-geometry
asked Sep 2 at 8:41
zoli
16.3k41643
The problem
Given the following figure
determine $x$.
My solution(s):
There are many possibilities to use similarities and the Pythagorean resulting more or less complicated systems of equations.
The simplest solution I could come up with was the following.
Because of the similarity of $AEF$ and $FCD$, we may say that $u=1/x$.
Based on the Pythagorean we have that
$$(1+x)^2+(1+1/x)^2=100.$$
This equation has two positive real solutions. One of them is:
$$x=frac12left(-1+sqrt101+sqrt98-2sqrt101right)approx 8.937.$$ The other one is $frac18.937.$
(http://www.wolframalpha.com/input/?i=solve+(1%2Bx)%5E2%2B(1%2B1%2Fx)%5E2%3D100)
The question
The source of the problem above is an old (Hungarian) high scholl level problem book. So, I suspect that there must be a simpler solution not requiring the roots of a fourth order equations. Please, either prove that there is no simpler solution or show one.
geometry euclidean-geometry
geometry euclidean-geometry
asked Sep 2 at 8:41
zoli
16.3k41643
asked Sep 2 at 8:41
zoli
16.3k41643
edited Sep 2 at 9:49
asked Sep 2 at 8:41
zoli
16.3k41643
asked Sep 2 at 8:41
zoli
16.3k41643
asked Sep 2 at 8:41
zoli
16.3k41643
16.3k41643
Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46
add a comment |Â
Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46
Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
I write down my thoughts here. If not satisfying you requirements, please inform me and I would delete this.
When solving
$$
(1+x)^2 + left(1 + frac 1xright)^2 = 100,
$$
we first expand it to
$$
x^2 + frac 1x^2 + 2 + 2x +frac 2x = 100,
$$
then notice that
$$
x^2 +frac 1x^2 + 2 = left(x + frac 1xright)^2,
$$
we let $y = x + 1/x$, then
$$
y^2 + 2y = 100,
$$
which is easily solvable by the roots formula of quadratic equations. Then all knowledge we used is taught before college, and the question could be used as a high school level geometry exercise.
answered Sep 2 at 9:55
xbh
3,515320
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
I write down my thoughts here. If not satisfying you requirements, please inform me and I would delete this.
When solving
$$
(1+x)^2 + left(1 + frac 1xright)^2 = 100,
$$
we first expand it to
$$
x^2 + frac 1x^2 + 2 + 2x +frac 2x = 100,
$$
then notice that
$$
x^2 +frac 1x^2 + 2 = left(x + frac 1xright)^2,
$$
we let $y = x + 1/x$, then
$$
y^2 + 2y = 100,
$$
which is easily solvable by the roots formula of quadratic equations. Then all knowledge we used is taught before college, and the question could be used as a high school level geometry exercise.
answered Sep 2 at 9:55
xbh
3,515320
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
add a comment |Â
up vote
0
down vote
accepted
I write down my thoughts here. If not satisfying you requirements, please inform me and I would delete this.
When solving
$$
(1+x)^2 + left(1 + frac 1xright)^2 = 100,
$$
we first expand it to
$$
x^2 + frac 1x^2 + 2 + 2x +frac 2x = 100,
$$
then notice that
$$
x^2 +frac 1x^2 + 2 = left(x + frac 1xright)^2,
$$
we let $y = x + 1/x$, then
$$
y^2 + 2y = 100,
$$
which is easily solvable by the roots formula of quadratic equations. Then all knowledge we used is taught before college, and the question could be used as a high school level geometry exercise.
answered Sep 2 at 9:55
xbh
3,515320
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
I write down my thoughts here. If not satisfying you requirements, please inform me and I would delete this.
When solving
$$
(1+x)^2 + left(1 + frac 1xright)^2 = 100,
$$
we first expand it to
$$
x^2 + frac 1x^2 + 2 + 2x +frac 2x = 100,
$$
then notice that
$$
x^2 +frac 1x^2 + 2 = left(x + frac 1xright)^2,
$$
we let $y = x + 1/x$, then
$$
y^2 + 2y = 100,
$$
which is easily solvable by the roots formula of quadratic equations. Then all knowledge we used is taught before college, and the question could be used as a high school level geometry exercise.
answered Sep 2 at 9:55
xbh
3,515320
I write down my thoughts here. If not satisfying you requirements, please inform me and I would delete this.
When solving
$$
(1+x)^2 + left(1 + frac 1xright)^2 = 100,
$$
we first expand it to
$$
x^2 + frac 1x^2 + 2 + 2x +frac 2x = 100,
$$
then notice that
$$
x^2 +frac 1x^2 + 2 = left(x + frac 1xright)^2,
$$
we let $y = x + 1/x$, then
$$
y^2 + 2y = 100,
$$
which is easily solvable by the roots formula of quadratic equations. Then all knowledge we used is taught before college, and the question could be used as a high school level geometry exercise.
answered Sep 2 at 9:55
xbh
3,515320
answered Sep 2 at 9:55
xbh
3,515320
answered Sep 2 at 9:55
xbh
3,515320
answered Sep 2 at 9:55
xbh
3,515320
3,515320
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
add a comment |Â
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
Shame on me. Thank you. This is clear.
– zoli
Sep 2 at 10:12
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
You are welcome. No shame here. Sometimes we may neglect something which is totally not a big deal.
– xbh
Sep 2 at 10:13
add a comment |Â
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Let $y=x+1/x$. Solve $y$ first, which is doable by formula of roots of quadratic equations.
– xbh
Sep 2 at 8:51
Do you mean that I should try the following two substitutions: $x_1,2=fracypm sqrty^2-42 ?$
– zoli
Sep 2 at 9:46