Simple Geometry Shows When Two Shapes Are Equal

Gina the geometry student stayed up too late last night doing her homework and when she finally went to bed she was still full of cupcakes and compasses. This lead to a strange dream.

Gina was the judge of the Great Brownie bake off at Imaginary University, a school where students don't learn a lot of math. It was up to Gina to decide which team made the biggest brownie.

Team Alpha presented a rectangular brownie to the judges. The brownie was 16 inches long and 9 inches wide, according to Gina. The square brownie was 12 inches on each side for TeamBeta. The trouble started when that happened.

Team Alpha's captain said that their brownie was longer than yours. We are the winners because ours is larger.

The side of our square is longer than the side of yours. The square is larger. We have accomplished something.

It was weird for Gina to argue about this. She said the area of the brownie is more than 100 square inches. The area of the brownie is 12 times larger than the square inch. The brownies are not different.

Patrick Honner is a high school teacher from New York.

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The teams looked confused. One student had never been taught multiplication. Another person said "Me neither." A third person said, "I heard about students at Complex College using numbers, but what does that mean?" Even as a dream goes, Imaginary University was a strange place.

Gina was asked what she was going to do. She would have to convince the teams that their brownies were the same size if they didn't know how to measure. Gina came up with a great idea. She asked if she could give her a knife.

Gina made a cut from the long side of the brownies to the short side. The large rectangular shape was turned into two smaller ones, measuring 9-by-12 and 9-by-4. She turned the 9-by-4 piece into three smaller ones. A bit of rearranging resulted in audible oohs and aahs from the crowd, as Gina made a replica of the square.

Both teams agreed that their brownies were the same size. Gina showed that the two brownies were the same size. Thousands of years ago, dissections like this were used to show that figures are the same size. Calculating when certain shapes are equivalent is still done using dissection and rearrangement.

Geometric dissections are a part of math class when it comes to basic shapes. If you recall, the area of a parallelogram is equal to the length of its base times its height.

The Great Brownie bake off was heating up. The team had a large brownie. They said that the winner was here. Both of our sides are very long.

Gina made a picture of the sides. She said that this has the same area. Gina paused for a second and noticed the puzzled looks on everyone's faces. Never mind. Give me the knife and I'll use it.

Gina sliced from the center of the hypotenuse to the center of the longer leg to form a triangle that would fit into the larger piece.

Team Alpha exclaimed, "That's our brownie!" The result was the same size as theirs.

The team had doubts. Their leader asked how the triangle compared to the square.

Gina was prepared for that. By transitivity, the triangle and the square are the same size. The most important property of equality is transitivity.

Gina was having a lot of fun with dissections. A few more cuts could be made.

Gina did a rotation of the triangle that was previously a triangle. She used the same pattern to cut it.

She showed how this new dissection of Team Alpha's triangle could be turned into TeamBeta's square.

The triangle and the square are both scissors that can be rearranged to form another. The brownies show how the scissors work.

The pattern can be used to turn a triangle into a square or a triangle into a square. scissors congruent to shape B are scissors congruent to shape A.

The triangle, the rectangle and the square are shown to be transitive. The triangle is scissors that are congruent to the square and the rectangles are scissors that are congruent to the square. The pattern is proof, just put it on the intermediate shape, like the one above.

scissors congruent is the result of the fact that two polygons have the same area. If you want to make the other one, you can either cut one up into a finite number of pieces or rearrange them to make the other.

The proof is very easy to prove. The first thing you need to do is slice the polygons into triangles.

Gina rearranged the triangular brownie by turning each triangle into a rectangular shape.

The tricky part is to turn each rectangular into a new one.

The pieces that need to be chopped off are from the same area.

Just stack them on top of each other if you can chop the rectangle into a number of pieces. Stack the rest on top of each other if the last piece is between 1 and 2 units wide.

If the rectangle is less than 1 unit wide, just slice it in half and use the two pieces to make a new one that is twice the length and half the thickness. Continue until you have a rectangular between 1 and 2 units.

Now imagine that this final rectangle has height h and width w, with 1 < w < 2. We’re going to cut up that rectangle and rearrange it into a rectangle with width 1 and height h × w. To do this, overlay the h × w rectangle with the desired hw × 1 rectangle like this.

The triangle at the bottom of the hw 1 rectangle should be cut from the dotted line.

The h w rectangle can be rearranged into 3 pieces. There are some clever arguments involving similar triangles that need to be made. For more information, see the exercises below.

If you put the last rectangle on top of the stack, you can turn it into a width 1

If the area of the original polygon was A, then the height of the rectangle must be A, so every one of them is scissors congruent to the other one. The scissors are congruent to each other in every other area.

The result wasn't enough to complete the judging of the brownie bake off Nobody was surprised by what Team Pi showed up with.

Gina woke up from her dream after seeing that circle. She knew that it wasn't possible to cut up a circle into many pieces and rearrange them to make a square or a rectangular shape. The mathematicians were able to prove that a circle isn't scissors congruent to anything. Gina's dream was turning into a nightmare.

Calculating this obstacle into new mathematics is what they do. As long as you can use infinitely small, infinitely disconnected, infinitely jagged pieces, it is possible to rearrange a circle into a square.

It proved that a decomposition like Laczkovich's is possible. It didn't explain how to build the pieces. In early 2022, Andras Mthé, Oleg Pikhurko and Jonathan Noel posted a paper in which they matched Laczkovich's accomplishment, but with pieces that can be seen.

They won't allow you to use their result to settle brownies. 10 200 pieces can only be produced by scissors. It's another step forward in answering a long line of questions related to the discovery of $latex pi$. New mathematics that previous generations couldn't dream of is kept moving.

Exercises

This is the first thing. Explain how the triangle we cut off fits into the space on the other side of the parallelogram.

There are two Explain why a triangle can be broken into two parts.

Consider the diagram used to show that an h w rectangle is scissors congruent to an hw 1 rectangle.

There are three. Why is $latex triangle$ XYQ similar to $latextriangle$ ABX? The length of QY is something to think about.

There are four. Explain why the PCX is congruent to thelatex.

Click for Answer 1:

The two triangles can be shown in many ways. The two right triangles have a pair ofcongruent legs because the distance between parallel lines is constant.

The two triangles are congruent in a parallelogram. The angle-side-angle triangle congruence theorem can be used to argue.

Click for Answer 2:

One of the best results in triangle geometry is the triangle midsegment theorem, which states that if you connect the middles of two sides of a triangle, the resulting line segment is half the length.

The segments are parallel to the third side. The angles are supplementary because they are the same side interior angles. The angles 3 and 2 are supplementary since $latexangle$ 1 is congruent to $latexangle$ 3.

The congruent sides will match up perfectly when you flip the top triangle around and to the right.

The triangle can be turned into a parallelogram if this is done.