- Identify and label three points on the coordinate plane that are a translation of the original triangle.

- Next, use the coordinates of your translation along with the distance formula to show that the two triangles are congruent by the SSS postulate.

- You must show all work with the distance formula and each corresponding pair of sides to receive full credit.

1.

Points of the new translated triangle are:

D=(16,0)

E=(8,0)

F=(16,6)

Triangle translated using the rule (x+10,y+3)

Proving congruence with the distance formula.

From these calculations we can gather that:

Line AB= Line DE

Line BC=Line EF

Line CA=Line FD

Therefore Triangle ABC is congruent to the translated Triangle DEF!

Step 2: Reflections and ASA

- Identify and label three points on the coordinate plane that are a reflection of the original triangle.

- Next, use the coordinates of your reflection to show that the two triangles are congruent by the ASA postulate.

- You can use the distance formula to show congruency for the sides. To show an angle is congruent to a corresponding angle, you can use slope or your compass and straightedge (Hint: Remember when you learned how to copy an angle?). You must show all work with the distance formula for the corresponding pair of sides and your work for the corresponding angles to receive full credit.

Points of the new reflected triangle are:

A'=(2,-3)

B'=(-6,-3)

C'=(-6,3)

Triangle was reflected over the y axis.

Proving congruence between Line AB and Line A'B'. I have proved the congruence of these sides using ASA.

To determine if lines AB and BC and A'B' and B'C' formed right angles in their triangles, I used the slope formula. I know that if the slope of Line B'C' is one number and the slope of Line BC is the reciprocal of that number as well as if the slope of Line A'B' is one number and Line AB's slope is the reciprocal of that number, both triangles have right angles. Therefore, those angles are congruent.

Line AB's slope is 0/8.

Line A'B' has a slope of 0/-8 which is flipped to remove the negative sign so it becomes 8/0.

8/0 is the reciprocal of 0/8.

Line BC's slope is 0/6.

Line B'C' has a slope of 6/0 which is the reciprocal of 0/6.

We have confirmed that Lines AB and BC on the original triangle and Lines A'B' and B'C' on the reflected triangle both form right angles and are therefore congruent. Now I have proved my first angle congruent using ASA.

Our last step is to prove the congruence of one more angle. I will use angles A and A'. I will prove their congruence using GeoGebra's Angle setting which will tell you how many degrees the angle measures.

As you can see, these two angles are congruent as they both measure 36.87 degrees.

IMPORTANT NOTE:

Using a computer drawing program is one way to get really accurate results very quickly, which is why this strategy is suggested in the How To Show Congruence with SSS, ASA, AND SAS Help For 2.06 Video which can be found in the Geometry Help website. The strategy is suggested exactly 6:37 seconds in the video as a way to prove the congruence of two angles for this assignment.

Step 3: Rotations and SAS

-Identify and label three points on the coordinate plane that are a rotation of the original triangle.

-Next, use the coordinates of your rotation to show that the two triangles are congruent by the SAS postulate. You can use the distance formula to show congruency for the sides. To show an angle is congruent to a corresponding angle, you can use slope or your compass and straightedge (Hint: Remember when you learned how to copy an angle?).

-You must show all work with the distance formula for the corresponding pair of sides and your work for the corresponding angles to receive full credit.

Rotated 180 degrees

Proof that Sides AB and A'B' are congruent as well as sides BC and B'C'.

Proof that both Lines AB and BC and Lines A'B' and B'C' form right angles on their triangles. Therefore these angles are congruent.

**These triangles are congruent!**

**These triangles are congruent!**

These triangles are congruent!

Reflection Questions

1. Describe the transformation you performed on the original triangle. Use details and coordinates to explain how the figure was transformed. Be sure to use complete sentences in your answer.

The transformation I performed on the original triangle was simply counting 10 spaces right and three spaces up from each of the three original points and placing a point there to form a new triangle. In Geometry, this rule is stated as (x+10, y+3). You can use this rule to come up with new coordinates even without a grid. Let's use point A on our original triangle for example. Point A's original coordinates are (6,-3). Add 10 to 6 and 3 to -3 and you get the coordinates to our new point (16,0). This is the original point's corresponding point. When you do this to the other two points as well, you have a new triangle that is congruent to the original! You can prove the two triangles are congruent using the distance formula.

2. How many degrees did you rotate your triangle? In which direction (clockwise, counterclockwise) did it move? Be sure to use complete sentences in your answer.

I rotated my triangle 180 degrees! It went clockwise.

3. What line of reflection did you choose for your transformation? How are you sure that each point was reflected across this line? Be sure to use complete sentences in your answer.

When I reflected the original triangle, I reflected it across the y axis. I know that all three points were reflected across the line because the image on the other side of the line matched the original. The new triangle was also proven congruent to the original!