Question

1. graph the ordered pairs to construct the triangles. a(-6,0), b(1,3), c(5,-3), d(-2,-6) draw a line between point a and point c. using the congruent triangle theorems, determine if \\( \triangle a b c \cong \triangle c d a \\).

Explanation:

Step1: Calculate the lengths of sides using the distance formula

The distance formula is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(AB\): \(x_1=-6,y_1 = 0,x_2 = 1,y_2=3\)
\(AB=\sqrt{(1+6)^2+(3 - 0)^2}=\sqrt{49 + 9}=\sqrt{58}\)
For \(BC\): \(x_1=1,y_1 = 3,x_2 = 5,y_2=-3\)
\(BC=\sqrt{(5 - 1)^2+(-3 - 3)^2}=\sqrt{16+36}=\sqrt{52}\)
For \(CD\): \(x_1=5,y_1=-3,x_2=-2,y_2=-6\)
\(CD=\sqrt{(-2 - 5)^2+(-6 + 3)^2}=\sqrt{49+9}=\sqrt{58}\)
For \(DA\): \(x_1=-2,y_1=-6,x_2=-6,y_2=0\)
\(DA=\sqrt{(-6 + 2)^2+(0 + 6)^2}=\sqrt{16 + 36}=\sqrt{52}\)
For \(AC\): \(x_1=-6,y_1 = 0,x_2 = 5,y_2=-3\)
\(AC=\sqrt{(5 + 6)^2+(-3 - 0)^2}=\sqrt{121+9}=\sqrt{130}\)

Step2: Use SSS (Side - Side - Side) congruence criterion

In \(\triangle ABC\) and \(\triangle CDA\)
\(AB = CD=\sqrt{58}\), \(BC=DA=\sqrt{52}\), \(AC = CA=\sqrt{130}\) (common side)

Answer:

Since \(AB = CD\), \(BC = DA\) and \(AC=CA\), by SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle CDA\)