Q. 28 - Geometry

Question

In Exercises 27-32 find and then compare lengths of segments.

 Quadrilateral TAUL has vertices T(4, 6), A(6, -4), U(~4, -2), and L(-2, 4), Show that the diagonals are congruent.

Step-by-Step Solution

Answer

The diagonals are congruent.

Step-1 – Given

The given vertices of the quadrilateral are $T\left(4,6\right),A\left(6,-4\right),U\left(-4,-2\right)\text{ and }L\left(-2,4\right)$.

Step-2 – To determine

We have to show that the diagonals of the quadrilateral are congruent.

Step-3 – Proof

The diagonals are TU and AL. We will use the distance formula to show TU = AL.

Using the distance formula, the length of TU is:

$\begin{array}{l}TU=\sqrt{{\left(-4-4\right)}^2+{\left(-2-6\right)}^2}\\ TU=\sqrt{{\left(-8\right)}^2+{\left(-8\right)}^2}\\ TU=\sqrt{64+64}\\ TU=\sqrt{128}\\ TU=8\sqrt{2}\end{array}$

And, the length of AL is:

$\begin{array}{l}AL=\sqrt{{\left(-2-6\right)}^2+{\left(4-\left(-4\right)\right)}^2}\\ AL=\sqrt{{\left(-2-6\right)}^2+{\left(4+4\right)}^2}\\ AL=\sqrt{{\left(-8\right)}^2+{\left(8\right)}^2}\\ AL=\sqrt{64+64}\\ AL=\sqrt{128}\\ AL=8\sqrt{2}\end{array}$

So, TU = AL so, the diagonals of the quadrilateral are congruent.