Here's the second diagram I taught at the 2016 PTN Workshops.

*I did not lay it out as I have done here. Today I think this diagram would have been a good handout.I could have drawn it; the participants could have followed along and had a cheat sheet to take home.*

4/18/2017: this diagram is awkward. I will redraw and simplify.

**Using the 3/4/5 triangle for construction**

**3/4 5 triangles always have a 90* angle where the side with 3 units meets the side with 4 units.**

Draw a line and mark off your unit.

Lay out lines of 3 units, 4 units and 5 units.

On my diagram: A-B = 3 units

A-C = 4 units

B-D = 5 units

Swing an arc from either end of A-B; one arc with a 4 unit radius, one arc with a 5 unit radius,

Where the arcs cross is E.

Draw lines from A to B to E to A.

This is a 3/4/5 triangle. The corner at A is 90*

For fun I have laid out another triangle beginning with 5 units, use 3 and 4 units for the radii of the arcs. It is another 3/4/5 triangle with a 90* corner.

We used Gunston Hall, built of brick by George Mason from 1755 to 1759, as an example. Mason was a real mason; he gave George Washington advice about mortar recipes. He would have used the 3/4/5 triangle when he built walls or, as a Master Mason, instructed others. The triangle was/is one way to keep brick square and true.

It would have been ordinary for him to use 3/4/5 geometry to design his house.

The base of the brick work at Gunston Hall is 4 units. The height of the brick work of the end wall at Gunston Hall measures 3 units. The diagonal is 5 units.

*The floor plan is also laid put using the 3/4/5 triangle. See my post for more information and drawings: http://www.jgrarchitect.com/2014/05/gunston-hall-ason-neck-virginia.html*

I asked the participants at the PTN session to divide the width of the Gunston Hall side elevation into 4 equal parts. I wanted them to draw the geometry for themselves, to see it come to life.

Again a handout with step by step instructions would have been helpful.

One of the first figures in the pattern books on Practical Geometry is the division of a line by a perpendicular. Here is

*Figure 5, Plate II,*of Asher Benjamin's

**The American Builder's Companion,**first edition published 1806.

Asher Benjamin's

*Figure 3, Plate II,*shows a

simply drawn 3/4/5 triangle expressed with units 6/8/10

with short arc lines at c, the top, to show the use of a compass to make a circle with the radius determined.

His description assumes a familiarity with the language of geometry and compasses.

"To make a perpendicular with a 10 foot rod. Let

*b a*be 6 feet; take eight feet in your compasses; from

*b*make the arch

*c*, with the distance ten feet from

*a*; make the intersection at

*c,*and draw the perpendicular,

*c b*. "