Chapter 4.4, Problem 2CYP

(a)

To determine

To graph: Both triangles on the same coordinate plane

Explanation of Solution

Given information: The given points are $J\left(2,5\right),K\left(1,1\right)\&L\left(5,2\right)$ and $N\left(-3,0\right),P\left(-7,1\right)\&Q\left(-4,4\right)$ .

Graph: The graph of both triangles JKL and NPQ are given below.

  

(b)

To determine

To explain: whether the triangles are congruent or not.

Answer to Problem 2CYP

Both triangles are congruent.

Explanation of Solution

Given information: The graph of the given vertices is as follows.

  

  

We can see that both triangles are congruent in the graph.

(c)

To determine

To write: The logical argument using coordinate geometry to support the conjecture of Part (b).

Answer to Problem 2CYP

Both triangles are congruent.

Explanation of Solution

Given information: The graph of the given vertices is as follows.

  

The given points are $J\left(2,5\right),K\left(1,1\right)\&L\left(5,2\right)$ and $N\left(-3,0\right),P\left(-7,1\right)\&Q\left(-4,4\right)$ .

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

By using distance formula.

  $\begin{array}{l}JK=\sqrt{{\left(1-2\right)}^2+{\left(1-5\right)}^2}\\ =\sqrt{1+16}\\ =\sqrt{17}\end{array}$$\begin{array}{l}NP=\sqrt{{\left(-7+3\right)}^2+{\left(1-0\right)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\end{array}$

  $\begin{array}{l}KL=\sqrt{{\left(5-1\right)}^2+{\left(2-1\right)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\end{array}$$\begin{array}{l}NQ=\sqrt{{\left(-4+3\right)}^2+{\left(4-0\right)}^2}\\ =\sqrt{1+16}\\ =\sqrt{17}\end{array}$

  $\begin{array}{l}JL=\sqrt{{\left(5-2\right)}^2+{\left(2-5\right)}^2}\\ =\sqrt{9+9}\\ =\sqrt{18}\end{array}$$\begin{array}{l}PQ=\sqrt{{\left(-4+7\right)}^2+{\left(4-1\right)}^2}\\ =\sqrt{9+9}\\ =\sqrt{18}\end{array}$

Since both triangles have equal sides to each other, therefore, $\Delta JKL\cong NPQ$