1. ## ancient Chinese math

Hello everyone,

Could someone please show me how to do this problem?

A sqaure walled city of unkown dimentions has 4 gates, 1 at the center of each side. A tree stands 20 paces from the north gate. One has to walk 14 paces southward from the south gate and then turn west and walk 1775 paces before he can see the tree. What are the dimensions of the city? Answer: Each side is 250 paces long

Solve the problem using a strategy that involves proportions

Would it be like 20+y+14=y-1775 or something like that?

Thank you very much

2. Is the tree inside the city or outside the city? I had assumed the person walking would see the tree through the gates; that is if the tree is inside the city.

EDIT:

Using the diagram attached.

Each tick mark represents a gate.

Let x be the length from the gate to the nearest vertex of the squared city wall.

The restriction for x is that it has to be greater than zero. Thus, x > 0.

Dimensions of the city is = 2x by 2x paces

AB = 20 paces
CD = 14 paces
DE = 1775 paces
BC = 2x
AD = AB + BC + CD = 2x + 20 + 14 = 2x + 34
BF = x
FG = BC + CD = 2x + 14
GE = DE - BF = 1775 - x

On the diagram there are 3 triangles:
$\displaystyle \triangle ABF$, $\displaystyle \triangle FGE$, and $\displaystyle \triangle ADE$

Using the concept of proportionality (similar triangles), you can find x. Each triangle have identical angles which add up to 180 degrees, but different sides. Thus, you have three ways to solve for x.

$\displaystyle \triangle ABF \sim \triangle FGE$

$\displaystyle {AB \over FG} = {BF \over GE}$

$\displaystyle {20 \over 2x + 14} = {x \over 1775 - x}$

or

$\displaystyle \triangle FGE \sim \triangle ADE$

$\displaystyle {FG \over AD} = {GE \over DE}$

$\displaystyle {2x + 14 \over 2x + 34} = {1775 -x \over 1775}$

or

$\displaystyle \triangle ABF \sim \triangle ADE$

$\displaystyle {AB \over AD} = {BF \over DE}$

$\displaystyle {20 \over 2x + 34} = {x \over 1775}$

Solving for x from each expression, you will obtain solutions of 125 and -142.

Obviously x cannot equal to -142 paces, because of the restriction placed on x. (x > 0)

Therefore, x = 125.

Finally, your dimensions of the city is:

2x by 2x = 2(125) by 2(125) = 250 paces by 250 paces

3. -I have assumed that tree is outside

-I have assumed that the tree is northward of north gate

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Watch the figure the triangle on the upper part is similar to the complete triangle (You can prove it)

Now if the triangles are similar the sides are in proportion use them