how do I prove that the diagonals of a rectangle are congruent using coordinate geometry?

Question

I have a rectangle on the x and y axis the bottom is on the (0,0) and one of the sides is on (0,b) then I have (a,b) then (a,0)

y
^
l P(0,b) Q(a,b)
l
l-------------------l
l l
l l
l____________l_________>x
O(0,0) R(a,0)

and imagine a dotted x in the middle of the rectangle I hope this helps it was the best I could do

Answer 1

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Now the 4 vertices of the rectangle OPQR as per your diagram in cyclic order are O (0,0), P (0,b), Q (a,b) and R (a,0).

This is the Distance formula of co-ordinate geometry:-
"The distance between 2 points A (j,k) and B (l,m) is given by:
Distance = Square root of [ (l - j)^2 + (m - k)^2 ]"

Using the above formula,

P (0,b), R (a,0).
PR = Square root of [ (a-0)^2 + (0-b)^2 ]
PR = Square root of ( a^2 + b^2 )

O (0,0), Q (a,b).
OQ = Square root of [ (a-0)^2 + (b-0)^2 ]
OQ = Square root of ( a^2 + b^2 )

Therefore, PR = OQ = √( a^2 + b^2 ) i.e Square root of ( a^2 + b^2 )


And, Hence, the diagonals of a rectangle are congruent.

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Answer 2

Show that the diagonals have equal lengths. Using the Theorem of Pythagoras,
(b - 0)^2 + (a - 0)^2 =? (a - 0)^2 + (b - 0)^2

Answer 3

Show that the length of the two diagonals is the same.

The first diagonal goes from (0,0) to (a,b). Its length is:
√{(a-0)² + (b-0)²} = √(a² + b²)

The second diagonal goes from (a,0) to (0,b). Its length is:
√{(0-a)² + (b-0)²} = √(a² + b²)

The lengths are the same so the diagonals are congruent.

Answer 4

You're trying to show that the distance from (0,0) to (a,b) is the same as the distance from (0,b) to (a,0).

You need to use the co-ordinate geometry formula for distance between points.

Square root of [ (difference between x values) squared + (difference between y values) squared]

Answer 5

hmmm...my brain does calculations like that automatically...let me think back to 8th grade geometry....hmmm....nope, still gonna let it be automated in my head...

The question is more difficult than the answer. Visualize it.

Answer 6

Sorry,but can you scan the problem because i can't understand like this

Answer 7

I only c your picture, and i can't answer any question? ;)