Chapter 4.4, Problem 2CYU

a.

To determine

Tograph:

The triangle on the same coordinates plane.

Answer to Problem 2CYU

The graph is as follows,

  

Explanation of Solution

Given information:

The triangle $△ABC$ has vertices $A\left(-3,-5\right),B\left(-1,-1\right),C\left(-1,-5\right).$

The triangle $△XYZ$ has vertices $X\left(5,-5\right),Y\left(3,-1\right),Z\left(3,-5\right).$

Graph:

  

Interpretation:

Graph points $A\left(-3,-5\right),B\left(-1,-1\right),C\left(-1,-5\right),X\left(5,-5\right),Y\left(3,-1\right),Z\left(3,-5\right)$ . Then graph the triangles $△ABC$ and $△XYZ$ by connecting their vertices.

b.

To determine

To explain:

Theconjecture for given condition by using graph.

Answer to Problem 2CYU

The conjecture for given condition by using graph is explained below.

Explanation of Solution

Given information:

Condition:

To whether the triangles are congruent.

The triangle $△ABC$ has vertices $A\left(-3,-5\right),B\left(-1,-1\right),C\left(-1,-5\right).$

The triangle $△XYZ$ has vertices $X\left(5,-5\right),Y\left(3,-1\right),Z\left(3,-5\right).$

  

As given graph, it is reflect $△ABC$ across the line $x=1$ then get $△XYZ$

To be precise, despite orientation, these two triangles have the same side lengths.

Therefore, $SSS$ postulate, so it is easily conclude that given triangles are congruent.

Therefore, conjecture will be, $△ABC\cong △XYZ$

c.

To determine

To calculate:

Logical argument using coordinate geometry.

Answer to Problem 2CYU

Three sides of $△ABC$ are congruent to three sides of $△XYZ$ and they are conjecture.

Explanation of Solution

Given information:

The triangle $△ABC$ has vertices $A\left(-3,-5\right),B\left(-1,-1\right),C\left(-1,-5\right).$

The triangle $△XYZ$ has vertices $X\left(5,-5\right),Y\left(3,-1\right),Z\left(3,-5\right).$

Calculation:

To support our conjecture, notice as have to prove that corresponding sides are congruent.

  $\begin{array}{l}\overline{AB}\cong \overline{XY}\\ \overline{BC}\cong \overline{YZ}\\ \overline{CA}\cong \overline{ZX}\end{array}$

So, let’s start:

Since as know the coordinates of the vertices, by calculating use the formula,

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Where,

  $\left(x_1,y_1\right)$ are coordinate of one endpoint and $\left(x_2,y_2\right)$ coordinates of another one.

Therefore,

Find the length of $\overline{AB}$

  $\begin{array}{l}AB=\sqrt{{\left(AC\right)}^2+{\left(BC\right)}^2}\\ =\sqrt{2^2+4^2}\\ =\sqrt{20}\\ =2\sqrt{5}\end{array}$

Find the length of $\overline{XY}$

  $\begin{array}{l}XY=\sqrt{{\left(XZ\right)}^2+{\left(ZY\right)}^2}\\ =\sqrt{2^2+4^2}\\ =\sqrt{20}\\ =2\sqrt{5}\end{array}$

Thus, three sides of $△ABC$ are congruent to three sides of $△XYZ$ .