# Geometry behind a painting: an exploration

*Hello Student!*

In this activity, you will explore how concepts of triangle congruency is presented on the *Geometric Abstraction 1*, a painting by *James Clarke*.
Two triangles are said to be congruent if it both has the same shape and size even if it shares a common side or it was reflected or rotated.
**Directions:**
In the applet below,

1. Use the *Reflect about line tool *to reflect triangle CDE, use line segment CD as the line of reflection.
· highlight triangle CDE then click side CD
2. Use the *Reflect about Point *tool to reflect triangle JLK. Use point F as the point reflection.
· highlight triangle JLK then click point F
3. Use the *Rotate around point *tool, highlight triangle ABI and input 180^{0},then click point F.
4. Use the *Distance or length *tool to measure
the side lengths of each triangle (click on the tool then click on a side of a
particular triangle you want to measure). Do this on all triangles and all
sides of a triangle.
Take note of your observations. Feel free to trace other triangles found on the geometric abstract using *Polygon *tool (You may also try reflecting/rotating those).

Once done, answer the questions below.

## Exploring triangles on the Geometric Abstraction

As you reflect/rotate the triangles, what have you observed?

After reflecting/rotating the triangles, are their corresponding triangles congruent?

What test of congruence can you use on the given triangles above?

What have you observed as you rotate triangle ABI 180^{0}?

What have you observed as you reflect triangle CDE?

**Directions:**In the applet below, the side lengths of each triangle are already shown. The triangle with lower opacity were the corresponding triangles constructed upon rotating/reflecting and the triangle with higher opacity were the original triangles. Note that these corresponding triangles were congruent. Feel free to move any vertex from the original triangles and observe what will happen on their corresponding triangles. You may also use the

*Angle tool*(click the tool then click inside of a triangle) to explore on their interior angles. Once done, answer the questions below.

## Expanding knowledge about congruent triangles

After exploring the triangles behind *Geometric Abstraction 1 by James Clark*, how can you describe the side lengths and interior angles of congruent triangles?

Describe how* James Clarke* used the concepts of triangle congruency on his geometric abstract.

## References:

*Clarke, J. (2016). Geometric abstraction 1. Retrieved from https://pixels.com/featured/geometric-abstraction-1-james-clarke.html
Congruent Triangles (2011). Math Open Reference. https://www.mathopenref.com/congruenttriangles.html
Lee, A. (n.d.). congruence. Retrieved from https://www.geogebra.org/m/qVy3qV8e*