Question

Now there is a square city of unknown size with a gate at the center of each side. There is a tree 20 b from the north gate. That tree can be seen when one walks 14 bu from the south gate, turns west and walks 1775 bu. Find the length of each side of the city.

Answer 1

The length of each side of the city is 250b

Explanation:

Given

`a = 20` --- tree distance from north gate

`b =14` --- movement from south gate

`c = 1775` --- movement in west direction from (b)

See attachment for illustration

Required

Find x

To do this, we have:

`\triangle ADE \sim \triangle ACB` --- similar triangles

So, we have the following equivalent ratios

`AE:DE = AB:CB`

Where:

`AE = 20\\ DE = x/2 \\ AB = 20 + x + 14 \\ CB = 1775`

Substitute these in the above equation

`20:x/2 = 20 + x + 14: 1775`

`20:x/2 = x + 34: 1775`

Express as fraction

`(20)/(x/2) = (x + 34)/(1775)`

`(40)/(x) = (x + 34)/(1775)`

Cross multiply

`x *(x + 34) = 1775 * 40`

Open bracket

`x^2 + 34x = 71000`

Rewrite as:

`x^2 + 34x - 71000 = 0`

Expand

`x^2 + 284x -250x - 71000 = 0`

Factorize

`x(x + 284) -250(x + 284)= 0`

Factor out x + 284

`(x - 250)(x + 284)= 0`

Split

`x - 250 = 0 \ or\ x + 284= 0`

Solve for x

`x = 250 \ or\ x =- 284`

x can't be negative;

So:

`x = 250`