Question

4.10 unit test: analytic geometry - part 1
due: thu nov 20, 2025 11:59pm
36 points possible
attempt 1
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complete the coordinate proof of the theorem.
given: abcd is a rectangle
prove: the diagonals of abcd are congruent
enter your answers in the boxes
the coordinates of rectangle abcd are a(0, 0), b(a, 0), c(□□), and d(0, b)

Explanation:

Step1: Analyze the properties of a rectangle

In a rectangle, opposite sides are equal and parallel. Since \(AB\) is along the \(x -\)axis from \(A(0,0)\) to \(B(a,0)\), the length of \(AB=a\). The side \(AD\) is along the \(y -\)axis from \(A(0,0)\) to \(D(0,b)\), so the length of \(AD = b\).
For the point \(C\), the \(x -\)coordinate is the same as the \(x -\)coordinate of \(B\) (because \(BC\) is parallel to \(AD\)) and the \(y -\)coordinate is the same as the \(y -\)coordinate of \(D\) (because \(CD\) is parallel to \(AB\)).

Step2: Determine the coordinates of \(C\)

The \(x -\)coordinate of \(C\) is \(a\) (matching the \(x -\)coordinate of \(B\)) and the \(y -\)coordinate of \(C\) is \(b\) (matching the \(y -\)coordinate of \(D\)). So the coordinates of \(C\) are \((a,b)\)

Answer:

\((a,b)\)