Question 1134281
the perimeter of the rectangle is 34 cm.


the formula for perimeter is 2x + 2y = p


when p = 34, the formula becomes 2x + 2y = 34


x is the length
y is the width


one of the diagonals is 13 cm.


since it's a rectangle, the other diagonal has to be 13 as well.


that's a nice fact to know but it's not necessary to know it to solve this problem.


the diagonal forms a right triangle with one of the sides being x and the other side being y.


by pythagorus, the diagonal squared is equal to x^2 + y^2.


the formula is therefore x^2 + y^2 = 13^2 which becomes x^2 + y^2 = 169.


you have two equations that needs to be solved simultaneously.


they are 2x + 2y = 34 and x^2 + y^2 = 169.


solve for y in the first equation to get y = 17 - x.


replace y with 17 - x in the second equation to get x^2 + (17 - x)^2 = 169


simplify to get x^2 + x^2 - 34x + 289 = 169


subtract 169 from both sides of that equation to get x^2 + x^2 - 34x + 120 = 0


combine like terms to get 2x^2 - 34x + 120 = 0


divide both sides of the equation by 2 to get x^2 - 17x + 60 = 0


factor that equation to get (x - 5) * (x - 12) = 0


that makes x = 5 or x = 12.


since y = 17 - x, .....


when x = 5, y = 12
when x = 12, y = 5


since x is the length of the rectangle, let x = 12.
you have length of the rectangle is 12 and width of the rectangle is 5.


the perimeter is equal to 2 * 12 + 2 * 5 = 24 + 10 = 35 cm which is correct.


x^2 + y^2 = 13^2 becomes 12^2 + 5^2 = 169 which becomes 144 + 25 = 169 which becomes 169 = 169 which is true.


the solution looks good.


the solution is that the length of the rectangle is 12 and the width of the rectangle is 5.


here's a good reference on the properties of a rectangle.


<a href = "https://brilliant.org/wiki/properties-of-rectangles/" target = "_blank">https://brilliant.org/wiki/properties-of-rectangles/</a>