Question

n: quadrilateral abcd with vertices a(-4, 3), b(1, 4), c(2, -1),... e: the diagonals of abcd are congruent. (14 points) i will use the distance formula to show... algebraic calculations: d=√(x2 - x1)² + (y2 - ...

Explanation:

Step1: Identify diagonals

The diagonals of quadrilateral \(ABCD\) are \(AC\) and \(BD\). For \(A(-4, 3)\) and \(C(2, -1)\); for \(B(1, 4)\) and \(D(-3, -2)\) (assuming \(D\) is \((-3, -2)\) from the graph).

Step2: Apply distance formula to \(AC\)

The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For \(A(-4, 3)\) and \(C(2, -1)\):
\(x_1=-4, y_1 = 3, x_2=2, y_2=-1\)
\(d_{AC}=\sqrt{(2 - (-4))^2 + (-1 - 3)^2}=\sqrt{(6)^2+(-4)^2}=\sqrt{36 + 16}=\sqrt{52}\)

Step3: Apply distance formula to \(BD\)

For \(B(1, 4)\) and \(D(-3, -2)\):
\(x_1 = 1, y_1=4, x_2=-3, y_2=-2\)
\(d_{BD}=\sqrt{(-3 - 1)^2 + (-2 - 4)^2}=\sqrt{(-4)^2+(-6)^2}=\sqrt{16 + 36}=\sqrt{52}\)

Step4: Compare distances

Since \(d_{AC}=\sqrt{52}\) and \(d_{BD}=\sqrt{52}\), the diagonals are congruent.

Answer:

The diagonals \(AC\) and \(BD\) have the same length (\(\sqrt{52}\)), so the diagonals of \(ABCD\) are congruent.