i know how to work this with actual numbers but writing the proof is freakin me out...here it is.
Prove analytically that the diagonals of a square are of equal length, bisect eachother, and are at right angles. help!
i know how to work this with actual numbers but writing the proof is freakin me out...here it is.
Prove analytically that the diagonals of a square are of equal length, bisect eachother, and are at right angles. help!
try to draw the figure so that you can it..
orient the square in the cartesian plane such that the vertices are of this form: W=(0,0), X=(a,0), Y=(a,a), Z=(0,a)
the diagonals are of equal length:
use pythagorean theorem.. $\displaystyle WY = \sqrt {a^2 + a^2} = XZ$, therefore, the diagonals are of equal in length..
diagonals bisect each other:
let us name the intersection of the diagonals, say E=(x,y)
Consider the triangles formed by YZE, ZWE, WXE and XYE: they are similar triangles. if they are similar triangles and at least one corresponding parts are equivalent, then the triangles are also equivalent.. so, the corresponding sides ZE, WE, XE and YE are equivalent which proves that they are of equal lengths, hence bisects each other (i.e. WY = WE + EY, and XZ = XE + EZ, and ZE = WE = XE = YE)..
forms right angles:
if we go back to the triangles, since ZE = WE = XE = YE, each triangles are isosceles right triangles, and so, opposite these equals sides are angles with $\displaystyle 45^0$ each. therefore, each angle formed in E would be 90..
Hello, the_rookie07!
I don't suppose you made a sketch . . .
Prove analytically that the diagonals of a square are of equal length,
bisect each other, and are at right angles.Code:| D(0,a)* - - - - - *C(a,a) | | | | | | | | | | - - * - - - - - * - - A(0,0) B(a,0)
Use the Distance Formula to compare the lengths of $\displaystyle AC$ and $\displaystyle BD.$
Write the equation of diagonals $\displaystyle AC$ and $\displaystyle BD$
. . and locate their intersection, $\displaystyle P.$
Then use the Distance Formula to show that: .$\displaystyle PA = PC$ and $\displaystyle PB = PD.$
You already have the equations of $\displaystyle AC$ and $\displaystyle BD.$
. . Simply show that they are perpendicular.