Sangaku Quest

Problem   If the diameter of the circle inscribed in the three sides of a right-angled triangle $ABC$ is $d$, and the length of the perpendicular $BD$ from the right angle $∠B$ to the hypotenuse $CA$ is known to be $a$, What are the lengths of the short side $AB$, the long side $BC$, and…

Problem If the differences between the hypotenuse $CA$ and the short side $AB$ of a right-angled triangle and between the hypotenuse $CA$ and the long side $BC$ are $a$ and $b$ respectively, what are the lengths of the short side, the long side, and the hypotenuse, and the diameter $d$ of the circle inscribed in…

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? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$…