Right Triangles
[1] Suppose there is a right triangle with a base of 3 chi and a height of 4 chi. What is the length of the hypotenuse?
Answer: 5 chi.
[2] Suppose there is a right triangle with a hypotenuse of 5 chi and a base of 3 chi. What is the length of the height?
Answer: 4 chi.
[3] Suppose there is a right triangle with a height of 4 chi and a hypotenuse of 5 chi. What is the length of the base?
Answer: 3 chi.
The Pythagorean theorem states: Square the base and the height, add them together, and take the square root of the sum to get the hypotenuse.
Also, square the height, subtract it from the square of the hypotenuse, and take the square root of the remainder to get the base.
Also, square the base, subtract it from the square of the hypotenuse, and take the square root of the remainder to get the height.
[4] Suppose there is a round log with a diameter of 2.5 chi. We want to make a square plank with a thickness of 7 cun. What is the width of the plank?
Answer: 2.4 chi.
Method: Square the diameter of 2.5 chi, subtract the square of 7 cun, and take the square root of the remainder to get the width.
[5] Suppose there is a tree 2 zhang tall. Its circumference is 3 chi. A vine grows at its base, wrapping around the tree 7 times and reaching the same height as the tree. What is the length of the vine?
Answer: 2 zhang and 9 chi.
Method: Multiply 7 times the circumference of 3 chi to get the height. The height of the tree is the base. Use these values to find the hypotenuse. The hypotenuse is the length of the vine.
[6] Suppose there is a square pond with sides of 1 zhang. A reed grows in the center, emerging 1 chi above the water. When pulled to the edge, the reed just reaches the bank. What is the depth of the water and the length of the reed?
Answer: The water depth is 1 zhang and 2 chi; the reed length is 1 zhang and 3 chi.
Method: Square half the side length of the pond, subtract the square of the reed's height above water (1 chi), and double the result. Divide this by twice the reed's height above water to get the water depth. Add the reed's height above water to get the reed's total length.
[7] Suppose there is a standing tree. A rope is tied to its top and touches the ground 3 chi away from the base. When the rope is pulled taut, it reaches 8 chi from the base and is fully extended. What is the length of the rope?
Answer: 1 zhang, 2 chi, and 1/6 of a chi.
Method: Square the distance from the base when pulled taut (8 chi). Divide this by the distance the rope touches the ground from the base (3 chi) and add 1 to the result. Add the distance the rope touches the ground from the base and divide by 2 to get the rope's length.
[8] Suppose there is a wall 1 zhang high. A pole is leaned against the wall, its top reaching the top of the wall. When the base of the pole is pulled 1 chi away from the wall, the top of the pole touches the ground. What is the length of the pole?
Answer: 5 zhang and 5 cun.
Method: Square the wall's height (10 chi). Divide this by the distance the pole's base is moved (1 chi). Add the distance the pole's base is moved and divide by 2 to get the pole's length.
[9] Suppose there is a round log buried in a wall, its size unknown. When chiseled to a depth of 1 cun, the chisel path is 1 chi long. What is the diameter of the log?
Answer: The log's diameter is 2 chi and 6 cun.
Method: Square half the chisel path's length. Divide this by the chisel's depth (1 cun) and add 1. Then, add the chisel's depth to get the log's diameter.
[10] Suppose a door is opened 1 chi from the doorstop but leaves a gap of 2 cun. What is the width of the door?
Answer: 1 zhang and 1 cun.
Method: Square the distance from the doorstop (1 chi). Divide this by half the gap (2 cun) and add 1 to the result. Then, add half the gap to get the door's width.
[11]Suppose a door's height is 6 chi and 8 cun greater than its width. The distance between its opposite corners is exactly 1 zhang. What are the height and width of the door?
Answer: The width is 2 chi and 8 cun; the height is 9 chi and 6 cun.
Method: Square the distance between the corners (1 zhang) to get the square of the diagonal. Take half the difference between the height and width, square it, double the result, and subtract it from the square of the diagonal. Take half of the remaining value and find its square root. Subtract half the difference between the height and width to get the door's width. Add half the difference between the height and width to get the door's height.
[12] Suppose there is a door of unknown height and width, and a pole of unknown length. When the pole is placed horizontally, it does not protrude more than 4 chi. When placed vertically, it does not protrude more than 2 chi. When placed diagonally, it fits perfectly. What are the height, width, and diagonal of the door?
Answer: The width is 6 chi, the height is 8 chi, and the diagonal is 1 zhang.
Method: Multiply the horizontal and vertical non-protruding lengths, double the result, and take the square root. Add the vertical non-protruding length to get the door's width, add the horizontal non-protruding length to get the door's height, and add both non-protruding lengths to get the door's diagonal.
[13] Suppose there is a bamboo stalk 1 zhang tall. Its top is broken and touches the ground 3 chi from the base. What is the height of the broken part?
Answer: 4 chi and 11/20 of a chi.
Method: Square the distance from the base where the top touches the ground (3 chi). Divide this by the bamboo's height (1 zhang) and subtract the result from the bamboo's height. Finally, divide the remainder by 2 to get the height of the broken part.
[14] Suppose two people, A and B, are standing at the same spot. A walks south at a speed of 7 units, while B walks east at a speed of 3 units. After walking 10 steps south, A turns northeast and meets B. What are the distances traveled by A and B?
Answer: B walks 10.5 steps east; A walks 14.5 steps to meet B.
Method: Square A's speed (7) and B's speed (3), add them together, and divide by 2 to get A's northeast speed. Subtract A's northeast speed from the square of A's initial speed (7) to get A's south speed. Multiply B's speed (3) by A's initial speed (7) to get B's east speed. Set A's south distance as 10 steps and multiply it by A's northeast speed. Also, set A's south distance as 10 steps and multiply it by B's east speed. These two products represent the distances traveled. Divide each distance by A's south speed to get the number of steps.
[15] Suppose there is a right triangle with a base of 5 bu and a height of 12 bu. What is the side length of the largest square that can fit inside the triangle?
Answer: 3 bu and 9/17 of a bu.
Method: Add the base and height to get the divisor. Multiply the base and height to get the dividend. Divide the dividend by the divisor to get the side length of the square.
[16] Suppose there is a right triangle with a base of 8 bu and a height of 15 bu. What is the diameter of the largest circle that can fit inside the triangle?
Answer: 6 bu.
Method: The base is 8 bu and the height is 15 bu. Use these values to find the hypotenuse. Add the base, height, and hypotenuse to get the divisor. Multiply the base and height, and double the result to get the dividend. Divide the dividend by the divisor to get the diameter of the circle.
[17] Suppose there is a square city with sides of 200 bu, with a gate at the center of each side. There is a tree 15 bu east of the east gate. How many bu south of the south gate should one walk to see the tree?
Answer: 666 and a half bu.
Method: The distance east of the east gate is the divisor. Square half the city's side length to get the dividend. Divide the dividend by the divisor to get the distance south of the south gate.
[18] Suppose there is a city 7 li wide (east to west) and 9 li long (north to south), with a gate at the center of each side. There is a tree 15 li east of the east gate. How many bu south of the south gate should one walk to see the tree?
Answer: 315 bu.
Method: Multiply the distance from the east gate to the southeast corner by the distance from the south gate to the southeast corner to get the dividend. The distance from the tree to the east gate is the divisor. Divide the dividend by the divisor to get the answer.
[19] Suppose there is a square city of unknown size, with a gate at the center of each side. There is a tree 30 bu north of the north gate. Walking 750 bu west from the west gate, one can see the tree. What is the side length of the city?
Answer: 1 li.
Method: Multiply the distances from the north and west gates to the tree, then multiply the result by 4 to get the square of the city's side length. Take the square root to get the side length.
[20] Suppose there is a square city of unknown size, with a gate at the center of each side. There is a tree 20 bu north of the north gate. Walking 14 bu south from the south gate and then 1,775 bu west, one can see the tree. What is the side length of the city?
Answer: 250 bu.
Method: Multiply the distance north of the north gate by the distance west, and double the result to get the dividend. Add the distance south of the south gate to get the divisor. Take the square root of the dividend divided by the divisor to get the city's side length.
[21] Suppose there is a square city with sides of 10 li, with a gate at the center of each side. A and B both start from the city center. B walks east, while A walks south for an unknown distance before turning northeast. They meet at the city wall. A's speed is 5 units, and B's speed is 3 units. What are the distances traveled by A and B?
Answer: A walks 800 bu south from the south gate and then 4,887.5 bu northeast to meet B. B walks 4,312.5 bu east.
Method: Square A's speed (5) and B's speed (3), add them together, and divide by 2 to get A's northeast speed. Subtract A's northeast speed from the square of A's initial speed (5) to get A's south speed. Multiply B's speed (3) by A's initial speed (5) to get B's east speed. Multiply half the city's side length by A's south speed and divide by B's east speed to get the distance A walks south from the south gate. Add half the city's side length to get the total distance A walks south. To find the hypotenuse (A's northeast path), multiply A's south distance by A's northeast speed. To find B's east distance, multiply A's south distance by B's east speed. Divide each distance by A's south speed to get the number of steps.
[22] Suppose there is a tree at an unknown distance. Four markers are placed 1 zhang apart, with the left two markers aligned with the tree. Looking from the rear right marker, the tree appears to be 3 cun inside the front right marker. What is the distance from the observer to the tree?
Answer: 33 zhang, 3 chi, 3 cun, and a half cun less.
Method: Square the distance between the markers (1 zhang) to get the dividend. The apparent shift of the tree (3 cun) is the divisor. Divide the dividend by the divisor to get the distance.
[23] Suppose there is a mountain west of a tree, its height unknown. The mountain is 53 li from the tree, and the tree is 9 zhang and 5 chi tall. A person standing 3 li east of the tree observes that the top of the tree is aligned with the mountain peak. The person's eye level is 7 chi above the ground. What is the height of the mountain?
Answer: 164 zhang, 9 chi, 6 cun, and a half cun.
Method: Subtract the person's eye level (7 chi) from the tree's height. Multiply the result by the distance between the tree and the mountain (53 li) to get the dividend. The distance between the person and the tree (3 li) is the divisor. Divide the dividend by the divisor and add the tree's height to get the mountain's height.
[24] Suppose there is a well with a diameter of 5 chi, its depth unknown. A 5-chi-long pole is placed vertically in the well. Looking from the top of the pole to the water's edge, the line of sight intersects the well wall 4 cun below the top. What is the depth of the well?
Answer: 5 zhang, 7 chi, and 5 cun.
Method: Subtract the line of sight intersection point (4 cun) from the well's diameter (5 chi). Multiply the result by the pole's length (5 chi) to get the dividend. The line of sight intersection point (4 cun) is the divisor. Divide the dividend by the divisor to get the well's depth.