Question

1. graph the triangle with vertices ( d(2,0) ), ( e(2,4) ), and ( f(6,2) ). then graph a triangle congruent to ( \triangle def ).

Explanation:

Step1: Plot original triangle vertices

For \(D(2,0)\): move 2 units right on \(x\)-axis, 0 units up on \(y\)-axis.
For \(E(2,4)\): move 2 units right on \(x\)-axis, 4 units up on \(y\)-axis.
For \(F(6,2)\): move 6 units right on \(x\)-axis, 2 units up on \(y\)-axis.
Connect \(D - E - F - D\).

Step2: Use translation for congruent triangle

Translation rule \((x,y)\to(x + a,y + b)\). Here \(a = 3\), \(b=-1\).
Calculate new vertices:
For \(D\): \(x=2+3 = 5\), \(y=0+( - 1)=-1\)
For \(E\): \(x=2 + 3=5\), \(y=4+( - 1)=3\)
For \(F\): \(x=6+3 = 9\), \(y=2+( - 1)=1\)
Plot \(D'(5,-1)\), \(E'(5,3)\), \(F'(9,1)\) and connect \(D' - E' - F' - D'\). Congruent triangles have the same side - lengths and angles. A translation (a rigid transformation) preserves side - lengths and angles.

Answer:

To graph \(\triangle DEF\) with vertices \(D(2,0)\), \(E(2,4)\), and \(F(6,2)\):

1. Locate point \(D\) at \(x = 2\), \(y=0\).
2. Locate point \(E\) at \(x = 2\), \(y = 4\).
3. Locate point \(F\) at \(x=6\), \(y = 2\).
4. Connect the points \(D\), \(E\), and \(F\) to form \(\triangle DEF\).

To graph a congruent triangle, we can use a translation (for example). Let's translate \(\triangle DEF\) 3 units to the right and 1 unit down.
The new vertices will be:

• \(D'(2 + 3,0-1)=(5,-1)\)
• \(E'(2+ 3,4 - 1)=(5,3)\)
• \(F'(6+3,2 - 1)=(9,1)\)

Locate \(D'\), \(E'\), \(F'\) and connect them.