Question

Complete the coordinate proof of the theorem. Given: A B C D is a rectangle. Prove: The diagonals of A B C D are congruent.

Art: A rectangle is graphed on a coordinate plane. The horizontal x-axis and vertical y-axis are solid. The vertex labeled as A lies on begin ordered pair 0 comma 0 end ordered pair. The vertex labeled as B lies on begin ordered pair a comma 0 end ordered pair. The vertex labeled as D lies on begin ordered pair 0 comma b end ordered pair. The vertex C is unlabeled. Diagonals A C and B D are drawn by dotted lines.

The coordinates of rectangle ABCD are A(0, 0), B(a, 0), C( , ), and D(0, b). The length of AC⎯⎯⎯⎯⎯ is equal to . The length of BD⎯⎯⎯⎯⎯ is equal to . The diagonals of the rectangle have the same length. Therefore, AC⎯⎯⎯⎯⎯ is congruent to BD⎯⎯⎯⎯⎯.

Answer 1

The diagonals AC and BD of rectangle ABCD are congruent. The length of AC is equal to BD. The length of BD is equal to AC.

Step-by-step explanation:

In a rectangle, opposite sides are parallel, and adjacent sides are perpendicular. Let's consider the coordinates of the vertices A(0, 0), B(a, 0), C (x, y), and D (0, b). Since ABCD is a rectangle, we know that AB is parallel to CD and BC is parallel to AD, and AB is perpendicular to BC and AD.

The length of AC can be calculated using the distance formula:

AC =
`√( (x - 0)^2 + (y - 0)^2 ) = √( x^2 + y^2 )`

Similarly, the length of BD can be calculated as:

BD =
`√( (0 - a)^2 + (b - 0)^2 ) = √( a^2 + b^2 )`

Now, since AC and BD are both diagonals of the rectangle, if we can show that
`\( √( x^2 + y^2 )` =
`√( a^2 + b^2 ) \)`, then the diagonals are congruent.

Consider the rectangle's properties: AB and CD are parallel, so a = x, and AD and BC are parallel, so b = y. Therefore,
`\( √( x^2 + y^2 ) = √( a^2 + b^2 ) \)` simplifies to
`\( √( a^2 + b^2 ) = √( a^2 + b^2 ) \)`, confirming that the diagonals AC and BD are indeed congruent.

Answer 2

Step-by-step explanation:

Given coordinates of the rectangle ABCD:

• A(0, 0), B(a, 0), C(x, y), D(0, b)

Lets first find coordinates of C

We can see that A and B are on same line horizontally, so C and D will be on the parallel line

The distance AB and CD are equal also distance AD and BC are equal, therefore

• AB = a - 0 = a, CD = a
• AD = b - 0 = b, BC = b

So the coordinates of C are:

• x = a, y = b

Now the diagonals:

• AC = √(a-0)² + (b-0)² = √a²+b²
• BD =√(0-a)² + (b -0)² = √a² + b²

Since AC = BD, we can state they are congruent