Japan “Sangaku” Research Institute

Problem If the length of the long side of a right triangle is $a$ and the diameter of the circle inscribed in the three sides is $d$, what are the lengths of the short side $x$ and the hypotenuse $y$ respectively? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem If the length of the short side of a right triangle is $a$ and the diameter of the circle inscribed in the three sides is $d$, what are the lengths of the long side $x$ and the hypotenuse $y$ respectively? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$…

Problem When the area of a right-angled triangle is $S$, and the diameter of a circle inscribed on the three sides is $d$, find the short side, long side, and hypotenuse of the right-angled triangle.　(In the figure below, $AB$ is the short side, $BC$ is the long side, and $CA$ is the hypotenuse.) $$ $$…

Problem If the length of the hypotenuse of a right-angled triangle is $a$ and the diameter of the circle inscribed in the right-angled triangle is $d$, what are the lengths of the short and long sides of the right-angled triangle?　(In the figure below, $AB$ is the short side and $BC$ is the long side.) $$…

Problem Find the diameter of a circle inscribed in three sides of a right triangle $ABC$ ($\angle B$ is a right angle). $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the lengths…

Problem Find the length of one side of a cube with volume 4361420.8 $m^3$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Prime factorization of 43614208 yields $$43614208=2^{15}*11^3,$$ $$∴ 42614208=(2^5×11)^3.$$ Therefore, the length…

Problem Find the length of a side of a square with an area of 542726.89 $m^2$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Prime factorization of 54272689 yields $$54272689=53^2×139^2.$$ So the side…

Problem I have a debt of 4.50 $yen$ now. It was promised to pay 1.50 $yen$ a year over three years, but suddenly it was decided to pay once in the first year. How much should I pay when the annual interest rate is 0.15? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$…

Problem There are two sections with different heights, and the areas of the upper and lower sections are 155 $m^2$ and 175 $m^2$ respectively. The height difference between the upper and lower sections is 1.5 $m$. The soil obtained by digging up 3 $m$ in an area of 20 $m^2$ is brought from the outside…

Hachiman Wake Shrine is located 4.6 km northwest of Kumayama Station on the JR Sanyo Main Line. Problem There are three sections (of different heights), upper, middle and lower, in the rice field. The areas of the upper, middle and lower sections are 210, 150 and 240$m^2$ respectively. The height differences between the upper and…