In Japan, the Common Test for University Admissions is administered every year to determine the level of achievement in basic studies at the upper secondary school level for those applying for admission to Japanese universities.
I look forward to solving the math problems in these every year with paper and pencil.
Wolfram Language version 12 began to introduce a computable version of Euclid-style synthetic geometry. I loved it and made a post 4 years ago proving some geometry theorems.
In version 14, it is now possible to find the values of a geometric quantities. Now I can solve the geometry problems on the test with Wolfram Language, I will try to solve them specifically.
Problems
Problems
There is a circle O with center at point O and radius 5. On the circumference of this circle, take two points A and B so that AB = 6. Also, take a point C on the circumference of circle O that is different from the two points A and B.
(i) How many is Sin of angle ACB? If angle ACB is obtuse, how many is its Cos?
(ii) Take point C such that angle ACB is obtuse and BC = 5. How long is AC?
(iii) Take point C such that the area of triangle ABC is maximized. When a line perpendicular to line AB is drawn from point C and its intersection with line AB is D, how many is the Tan of angle OAD? Also, how many is the area of triangle ABC?
(iv) Take point C as in (iii) so that C has the largest area of triangle ABC. In this case, how many is the Tan of angle ACB?
Furthermore, draw a line perpendicular to line AC through point C and let E be the intersection with line AB. In this case, how many is Sin of angle BCE?
When taking point F on line CE, what is the minimum length of BF?
(i) How many is Sin of angle ACB? If angle ACB is obtuse, how many is its Cos?
(ii) Take point C such that angle ACB is obtuse and BC = 5. How long is AC?
(iii) Take point C such that the area of triangle ABC is maximized. When a line perpendicular to line AB is drawn from point C and its intersection with line AB is D, how many is the Tan of angle OAD? Also, how many is the area of triangle ABC?
(iv) Take point C as in (iii) so that C has the largest area of triangle ABC. In this case, how many is the Tan of angle ACB?
Furthermore, draw a line perpendicular to line AC through point C and let E be the intersection with line AB. In this case, how many is Sin of angle BCE?
When taking point F on line CE, what is the minimum length of BF?
Solutions
Solutions
There is a circle O with center at point O and radius 5. On the circumference of this circle, take two points A and B so that AB = 6. Also, take a point C on the circumference of circle O that is different from the two points A and B.
(i) How many is Sin of angle ACB? If angle ACB is obtuse, how many is its Cos?
(i) How many is Sin of angle ACB? If angle ACB is obtuse, how many is its Cos?
We only need to write the code as the problems states to get the diagram and the correct answer.
In[]:=
scene=GeometricScene[{o,a,b,c},{CircleThrough[{a,b,c},o,5],EuclideanDistance[a,b]6,Style[PlanarAngle[{a,c,b}],Orange]>90°}];RandomInstance[scene]
GeometricSolveValues[scene,Sin[PlanarAngle[{a,c,b}]]]
Out[]=
3
5
In[]:=
GeometricSolveValues[scene,Cos[PlanarAngle[{a,c,b}]]]
Out[]=
-
4
5
(ii) Take point C such that angle ACB is obtuse and BC = 5. How long is AC?
This problem can also be solved by simply writing the code as the problem.
In[]:=
scene=GeometricScene[{o,a,b,c},{CircleThrough[{a,b,c},o,5],EuclideanDistance[a,b]6,PlanarAngle[{a,c,b}]>90°,EuclideanDistance[b,c]5,Style[Line[{a,c}],Orange]}];RandomInstance[scene]
In[]:=
GeometricSolveValues[scene,EuclideanDistance[a,c]]
Out[]=
{-4+3
3
}(iii) Take point C such that the area of triangle ABC is maximized. When a line perpendicular to line AB is drawn from point C and its intersection with line AB is D, how many is the Tan of angle OAD? Also, how many is the area of triangle ABC?
I could not figure out how to describe "the area of triangle ABC is maximal" as stated in the problem statement. So I wrote the codes with a yellow background.
In[]:=
scene=GeometricScene[{o,a,b,c,d},{CircleThrough[{a,b,c},o,5],EuclideanDistance[a,b]6,
PlanarAngle[{a,c,b}]<90°
,EuclideanDistance[a,c]EuclideanDistance[b,c]
,d∈InfiniteLine[{a,b}],PlanarAngle[{a,d,c}]==90°,p==Polygon[{a,b,c}],Style[PlanarAngle[{o,a,d}],Orange]}];RandomInstance[scene]
In[]:=
GeometricSolveValues[scene,Tan[PlanarAngle[{o,a,d}]]]
Out[]=
4
3
In[]:=
GeometricSolveValues[scene,Area[p]]
Out[]=
{27}
(iv) Take point C as in (iii) so that C has the largest area of triangle ABC. In this case, how many is the Tan of angle ACB?
In[]:=
scene=GeometricScene[{o,a,b,c,d},{CircleThrough[{a,b,c},o,5],EuclideanDistance[a,b]6,PlanarAngle[{a,c,b}]<90°,EuclideanDistance[a,c]EuclideanDistance[b,c],d∈InfiniteLine[{a,b}],PlanarAngle[{a,d,c}]==90°,p==Polygon[{a,b,c}],Style[PlanarAngle[{a,c,b}],Orange]}];RandomInstance[scene]
Furthermore, draw a line perpendicular to line AC through point C and let E be the intersection with line AB. In this case, how many is Sin of angle BCE?
When taking point F on line CE, what is the minimum length of BF?
I could not figure out how to describe “the minimum length of BF" as stated in the problem. So I wrote the code with a yellow background.
Conclusion
Conclusion
The introduction of a computable version of Euclid-style synthetic geometry in version 12 is a wonderful feature. It gives us the illustrations we need when solving geometry problems. Version 14 takes it a step further and allows us to find the values of a geometric quantities.
In the future, we expect to be able to directly code(yellow background) the problem statement described in Solutions section. If anyone already knows of a way to express this using GeometricAssertion, I would appreciate it if you could let me know.
In addition, with the Wolfram Plugin, it will soon be possible to solve these problems by simply inputting the problem text directly into ChatGPT.
The output at this time of inputting the above problem into ChatGPT is as follows
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Solving the Common Test for Japanese Universities using Wolfram Language
by Kotaro Okazaki
Wolfram Community, STAFF PICKS, January 16, 2024
https://community.wolfram.com/groups/-/m/t/3103450
by Kotaro Okazaki
Wolfram Community, STAFF PICKS, January 16, 2024
https://community.wolfram.com/groups/-/m/t/3103450