## Posts tagged ‘isosceles’

### The Amazing National Flag of Nepal

The National Flag of Nepal is unique.

Here are several trivia questions about the Nepalese flag:

1. It is the only non-quadrilateral national flag in the world. What is its shape?
2. It is one of only three national flags where the height is not less than the length. What are the other two?
3. What is the sum of the three acute interior angles within the flag?

Question 3 may be difficult to answer without knowing more about the exact dimensions of the flag. For help with that, we turn to the Constitution of Nepal, promulgated 20 September 2015, which contains the following geometric description for the construction of the flag:

SCHEDULE – 1
The method of making the National Flag of Nepal

1. Method of making the shape inside the border
1. On the lower portion of a crimson cloth draw a line AB of the required length from left to right.
2. From A draw a line AC perpendicular to AB making AC equal to AB plus one third AB. From AC mark off D making the line AD equal to line AB. Join B and D.
3. From BD mark off E making BE equal to AB.
4. Touching E draw a line FG, starting from the point F on line AC, parallel to AB to the right hand-side. Mark off FG equal to AB.
5. Join C and G.

The traditionalist in me wishes that “line segment” were used instead of “line,” or that the overline were used to indicate those segments, and that a few more commas were inserted to make it more readable. Consequently, the math editor in me feels compelled to rewrite the directions as follows:

But the American in me — given how many times someone in the United States has tried to legislate the value of π — well, I’m just excited to see accurate mathematics within a government document.

The description continues for another 19 exhilarating steps, explaining how to construct a crescent moon in the top triangle, a twelve-pointed sun in the bottom triangle, and a border around the shape described above. Those steps are omitted here — because you surely get the gist from what’s above — but the following “explanation” that appears below the method is worthy of examination:

The lines HI, RS, FE, ED, JG, OQ, JK and UV are imaginary. Similarly, the external and internal circles of the sun and the other arcs except the crescent moon are also imaginary. These are not shown on the flag.

The entirety of this construction, as any classical geometrician would hope, can be completed with compass and straightedge. I cheated a bit and used Geometer’s SketchPad, with this being the resultant mess:

Geometer’s SketchPad Construction of Nepalese Flag

The rough part was placing C so that AC = AB + 1/3 AB. Geometer’s SketchPad could have easily measured AB, calculated 4/3 of its length, and then constructed a “circle by center and radius,” but that felt like cheating. Instead, I…

• located Q, which is halfway between A and D;
• constructed circle A with radius AQ = AP;
• constructed circle P with radius PD;
• constructed circle D1 with radius DA;
• located the intersections of circle P and circle D1 at points X and Y;
• constructed a line through X and Y;
• located R, which is 1/3 of the way from D to A;
• constructed circle D2 with radius DR; and,
• located C, so that CD = 1/3 AB.

Now, you could use that information to determine CF and FG, and then use the arctan function to calculate the measures of the two acute angles in the upper pennon. If you were so inclined, you’d find that their measures are 32.06° and 57.94°, respectively.

But the question above asked for the sum of the three angle measures. Without any work at all, it’s clear that the sum of those two angles must be 90°, since the construction described above implies that ΔCFG is a right triangle.

And because AB = AD by construction, then ΔDAB is an isosceles right triangle, and the measure of the third acute angle must be 45°.

And that brings us to a good point for revealing the answers to the three questions from above.

1. Pentagon
2. The flags of Switzerland and Vatican City are square, so the height and width are equal.
3. 135°

If you’re looking for more flag-related fun, check out the MJ4MF post from Flag Day 2016 about converting each flag to a pie graph.

### 4 Folds, 40 Theorems, and Chinese New Year

If you asked a mathematician how to make an origami coin purse for Chinese New Year, you might receive the following instructions:

1. Rotate a square piece of paper 45°.
2. Reflect an isosceles right triangle about a diagonal of the square so that congruent isosceles right triangles coincide.
3. Reflect the lower left vertex of the isosceles right triangle so it coincides with the opposite leg, such that the base of the isosceles right triangle and one side of the newly formed isosceles acute triangle are parallel.
4. Reflect the other vertex of the isosceles right triangle to coincide with a vertex of the isosceles triangle formed in the previous step.
5. Fold down a smaller isosceles right triangle from the top.
6. Fill the coin purse with various denominations of currency, such as a \$2 bill, a \$1 gold coin, a quarter, or a penny.

Don’t like those directions? Perhaps these pictures will help.

To be truly authentic for Chinese New Year, you’ll want to use red paper instead of white.

The Common Core State Standards for Mathematics (CCSSM) have added rigor to geometry by requiring students to prove no fewer than a dozen different theorems. Further, the geometry progressions document formulated by Dr. Hung-Hsi Wu suggests that a high school geometry course should use a transformational approach, basing a majority of the foundational axioms on rotations, translations, and reflections.

What struck me about the origami coin purse was the myriad geometry concepts that could be investigated during this simple paper-folding activity. Certainly, the idea to use origami in geometry class in nothing new, but this one seems to have a particularly high return on investment. Creating the coin purse requires only four folds, and though perhaps I lied by claiming forty theorems in the title, the number of axioms, definitions, and theorems that can be explored is fairly high. (Claiming “forty theorems” wasn’t so much untruthful as much as it was archaic. The word forty has been used to represent a large but unknown quantity, especially in religious stories or folk tales, as in Ali Baba and the 40 Thieves, it rained “40 days and 40 nights,” or Moses and the Israelites wandered the desert for 40 years.)

For instance, the fold in Step 3 requires parallel lines, and the resulting figures can be used to demonstrate and prove that parallel lines cut by a transversal result in alternate interior angles that are congruent.

No fewer than four similar isosceles right triangles are formed, and many congruent isosceles right triangles result.

The folded triangle in the image above is an isosceles acute triangle; prove that it’s isosceles. And if you unfold it from its current location, the triangle’s previous location and the unfolded triangle together form a parallelogram; prove it.

The final configuration contains a pair of congruent right triangles; you can use side-angle-side to prove that they’re congruent.

But maybe you’re not into proving things. That’s all right. The following are all of the geometric terms that can be used when describing some facet of this magnificent creation.

• Parallel
• Congruent
• Similar
• Alternate
• Angle
• Square
• Reflection
• Rotation
• Translation
• Transversal
• Bisect
• Coincide
• Right Angle
• Perpendicular
• Isosceles
• Vertex
• Triangle
• Trapezoid
• Parallelogram
• Pentagon

And the following theorems and axioms can be proven or exemplified with this activity.

• Sum of angles of a triangle is  180°.
• Alternate interior angles formed by a transversal are congruent.
• Any quadrilateral with four congruent sides is a parallelogram.
• Congruent angles are formed by an angle bisector.
• Triangles with two sides and the included angle congruent are congruent.

There is certainly more to be found. What do you see?

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