4.2: 2-D Geometry

Two-dimensional figures

Definition:

Vertices (pl.) or vertex (sg.): a point or corner that joins two shape edges.

Edges: A line that describes one of the outer borders of a shape.

A curve is called a closed curve if we can trace it so that the starting and ending points are the same.

A simple closed curve is a curve that can be traced without passing through any point more than once, with the starting and ending points being the same.

Polygons

A polygon is a closed, 2-dimensional shape with straight edges (sides). The word “polygon” is derived from Greek for “many angles”. The names of the polygons are derived from Greek number prefixes followed by gon, with only a couple of exceptions.

Example \(\PageIndex{1}\):

3-tri, 4-tetra, 5-penta, 6-hexa, 7-hepta, 8-octa, 9-ennea or nona, 10-deca.

Definitions:

Convex polygons are polygons such that any line drawn across them intersects only two edges.

Concave polygons are those in which more than two edges are intersected by a line drawn across the shape.

A polygon is considered regular if all of its sides have equal length and all its interior angles are equal.

A diagonal is a line that joins any two vertices of a shape that are not adjacent to each other.

Two polygons are Congruent if they are identical in size and shape.

• Interior angles of a polygon are the angles between two adjacent sides on the inside of the polygon
• Exterior angles of a polygon are the angles between two adjacent sides on the outside of the polygon
• Central angles are angles between two points on a polygon, measured from the center of the polygon. In circles, this measure is given in radians or degrees.

The center of a regular polygon is a point that is the same distance (equidistant) from all of its vertices. An irregular polygon will not have a center.

The radius of a circle is given by the distance from the center to the edge.

Regular Polygons

Properties:

A regular polygon's apothem is a line that is drawn from its center to the midpoint of one of its edges.

A regular polygon's radius is a line that is drawn from its center to one of its vertices.

The perimeter of a shape is the sum of the lengths of all its edges.

The area of a polygon is the measure of its surface, given in square units such as cm².

Triangles

A triangle is a polygon with three edges and three vertices and can be defined by a side and two angles. This means that if we know the length of one side and the measure of two angles in a triangle, we can know the measures of all three sides and angles.

Definition:

• Equilateral triangle: A triangle in which all three sides are equal in length.
• Isosceles triangle: A triangle with two sides equal in length.
• Scalene triangle: A triangle with no sides equal in length.
• Right triangle: A triangle with one interior angle that measures 90°.
• Acute-angled triangle: All angles are acute (between 0° and 90°).
• Right-angled triangle: One angle is a right angle (90°).
• Obtuse-angled triangle: One angle is obtuse (between 90° and 180°).

Example \(\PageIndex{2}\): Classify Triangles

Complete the following table by sketching a triangle, if possible. Otherwise, justify your answer.

Table \(\PageIndex{1}\): Triangle classifications

	Isosceles Triangle	Equilateral Triangle	Scalene Triangle
Acute angled
Obtuse-angled
Right-angled

Would you be able to sketch all of the above triangles inside a rectangle? Here is an Euler diagram of types of triangles:

Special properties of triangles

Definition:

• Angle Bisector: A line that bisects an angle of a triangle.
• Median: A line segment that connects a vertex to the midpoint of the opposite side.
• Altitude: A perpendicular line segment from a vertex to the opposite side.

Example \(\PageIndex{3}\): Proof - The Sum of Interior Angles of Any Triangle

The sum of the interior angles of any triangle is 180°:

Lines \(AB\) and \(CD\) are parallel.

By the alternate interior angles property, \(\measuredangle AEC\) and \(\measuredangle ECD\) are equal.

By the alternate interior angles property, \(\measuredangle BED\) and \(\measuredangle EDC\) are also equal.

Since \(\measuredangle AEC + \measuredangle CED + \measuredangle BED = 180°,\)

Then \(\measuredangle ECD + \measuredangle CED + \measuredangle EDC = 180°\).

Example \(\PageIndex{4}\): Proof - \(a^2 + b^2 = c^2\), or the Pythagorean Theorem

Method 1: Let us draw two squares made by using four right triangles with sides \(A, \, B, \, C\), where \(A\) is the vertical side, \(B\) is the horizontal side, and \(C\) is the hypotenuse:

Since both squares are the same size, their areas are equal.

Both squares also contain four right triangles.

When we remove the triangles from both squares, we are left with \(c^2\) on the left and \(a^2 + b^2\) on the right.

Remember, we have two squares of equal area and removed the same area from each.

So: \(a^2 + b^2 = c^2\).

Method 2: President Garfield's method

Example \(\PageIndex{5}\): \(\sqrt{2}\) is Irrational

Example \(\PageIndex{6}\): Kepler Triangle

A Kepler Triangle is a right-angled triangle with side lengths \( 1, \sqrt{\phi}, \phi \), where \( \phi \) is the golden ratio.

Example \(\PageIndex{7}\):

Triangle inequality

\(x+y \geq z\).

The sum of the lengths of two sides of any triangle is always greater than or equal to the length of the third side.

Definition:

Two triangles are similar if corresponding angles are equal and corresponding side lengths
are proportional.  The sides of similar triangles are in relative proportion: the ratio by which one is bigger or smaller than the other is the same for all sides.

Triangles are congruent if their corresponding sides and angles are equal.

Quadrilaterals

Quadrilaterals are polygons with four edges and four vertices.

• A kite is a quadrilateral with adjacent pairs of equal sides.
• A trapezoid is a quadrilateral with one pair of parallel sides.
• A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
• A rhombus is a parallelogram with all four sides equal in length.
• A rectangle has four right angles; thus, its opposite sides are parallel and equal in length (a quadrilateral with 4 square corners).
• A square has four right angles and four equal sides. (A quadrilateral with 4 square corners and 4 equal sides. Thus, a square is a special type of rectangle!)

Here is an Euler diagram of types of quadrilaterals:

Thinking Out Loud:

Draw a quadrilateral. Find the midpoint of each side (paper folding might help). Connect the midpoints of the sides that have a common vertex. What shape have you created? Is this true for any quadrilateral?

Example \(\PageIndex{4}\): Proof - The sum of the interior angles of a quadrilateral is \(360^\circ\)

First, let's draw a quadrilateral:

Then we will decompose this shape into an aggregate of shapes we know better: triangles.

As we can see, a quadrilateral is made of two triangles. Since the sum of the internal angles of each triangle is \(180^\circ\), we can add the angles of the two triangles to find that the sum of the internal angles of a quadrilateral is \(360^\circ\).

Thinking Out Loud:

Does the proof of the interior angles of a quadrilateral hold when the quadrilateral is concave? Why? Why not?

Example \(\PageIndex{8}\):

Express the relationship among a rhombus, a square, and a rectangle using a Venn diagram.

Pentagrams

Pentagrams are polygons that make a five-pointed star.

Example \(\PageIndex{9}\): Proof: The sum of the vertex angles of a Pentagram is 180°

Consider the following pentagram:

As we have determined, we can find the interior angles of the black triangle in the general case by using the internal vertex angles of the blue and red triangles.

Since the sum of a triangle's interior angles is 180°,

\(a + b + c + d + e = 180°\).

Other Polygons

The sum of the interior angles of polygons with \(n\) sides is given by the formula \((n - 2)180°\). This can be shown by the fact that \(n\) sided polygons can be divided into \((n-2)\) triangles.

Example \(\PageIndex{10}\):

Consider a regular decagon: \(n = 10\). What is the sum of its internal angles?

Since the sum of interior angles \(= (n-2)180°\),

Then the sum of a regular decagon's interior angles \(= (10 - 2)180°\)

\(= (8)180°\)

\(= 1440°\)

Similarity

Definition

Polygons with \(n\) sides are similar when their corresponding sides are proportional.

Example \(\PageIndex{8}\):

Tessellation

Definition

Tessellation refers to the "tiling" of polygons, as in ceramic tiles or quilts. There are regular tessellations and semi-regular tessellations. A regular tessellation is made up of regular polygons that are all of the same shape, and all meet at a common vertex. There are only three types of regular tessellations, which are comprised of equilateral triangles, squares, and hexagons. The sum of the angles of the polygons at each vertex in a regular tessellation is 360 degrees.

 

Thinking Out Loud:

Given a 2-dimensional stage, which shapes can be tiled? Which can't? Why?

Given a 3-dimensional space, does your answer change? Why or why not?

Definition:

A semi-regular tessellation is made up of two or more types of regular polygons. Polygons are arranged in the same cyclic order. There are eight semi-regular tessellations which comprise different combinations of equilateral triangles, squares, hexagons, octagons, and dodecagons.

Example \(\PageIndex{11}\): Semi-regular tessellations.

Definition:

Non-Regular Tessellations

Non-regular Tessellations are those in which there are no restrictions regarding the shapes used or their arrangement around vertices. It is believed that there are infinitely many irregular tessellations.

Curved figures

• Circles
• Semicircle
• Spiral
• Parabola
• Ellipse
• Hyperbola
• Crescent

Circle Geometry

Definitions:

The center of a circle is a point equidistant from all points on the circle's edge. The radius gives this distance.

The radius of a circle is the distance from the center to the edge.

The diameter of a circle is the distance across a circle at its widest point and is twice the circle's radius.

The circumference of a circle is the distance around the circle.

A chord is a line that joins two points on the circumference of a circle.

An arc is a part of the circumference of a circle.

A tangent line meets a circle at one point only.

An inscribed angle is formed by two chords that share a common endpoint.

A central angle is an angle describing an arc. This is measured in degrees and radians.

Properties

1. The perpendicular line from the center of a circle to a chord bisects the chord.
2. \(\pi=\dfrac{ \text{Circumference}}{\text{Diameter}}\).
3. \(A = \pi r^2\)
4. The angle between the tangent at a point and the radius from the center to that point is \(90^{\circ}\).
5. The measure of a central angle is twice as large as the measure of an inscribed angle that cuts the same arc of a circle.
6. The angle in a semicircle is a right angle.

Thinking Out Loud:

Can we inscribe regular polygons in a paper-cut circle?

Coordinate Geometry

Coordinate geometry applies the Cartesian plane to the geometry we have already learned. Each vertex of a shape now is given an ordered Cartesian pair \((x, \, y)\) to give its position on the grid. We create the right triangles and apply the Pythagorean Theorem to find distances between points.

Example \(\PageIndex{10}\):

In a general case, point A has coordinates \((x_1, \, y_1)\), and point B has coordinates \((x_2, \, y_2)\). Point C, the point we use to create a right triangle, has coordinates \((x_2, \, y_1)\).

Let \(A (x_1, \, y_1)\), and \( B (x_2, \, y_2)\) be points.

The coordinates of a line segment's midpoint can be found by averaging the coordinates of its endpoints, \(A\) and \(B\). So, the formula for the midpoint of a line is:

\(\left( \displaystyle \frac{x_1 + x_2}{2}, \, \displaystyle \frac{y_1 + y_2}{2} \right)\)

The slope of a Line:

The slope of a line, or how much it rises for a given amount of horizontal travel, can be thought of as \(m = \displaystyle \frac{rise}{run} = \displaystyle \frac{\Delta y}{\Delta x}\).

\(\Delta\) is the mathematical symbol used to denote the change in a quantity.

Hence the slope of the line \(AB\) is given by \(m = \displaystyle \frac{y_2 - y_1}{x_2 - x_1}\).

Distance Formula

The distance between the points \(A\) and \(B\) is given by \(\sqrt{ y_2 - y_1)^2+(x_2 - x_1)^2}\).

Example \(\PageIndex{12}\):

By using coordinate geometry, show that the line connecting the midpoints of two sides of a triangle is parallel to the other side.

First, let's draw a picture:

Let us call the midpoints of lines \(AB\) and \(BC\) \(p\) and \(q\), respectively.

Consider the slope of line \(AC\):

\(m\)_\(AC\)\(= \displaystyle \frac{y_c - y_a}{x_c - x_a}\)

Consider point \(p\), the midpoint of line \(AC\):

\(p = \left( \displaystyle \frac{x_a + x_b}{2}, \, \displaystyle \frac{y_a + y_b}{2} \right)\)

Consider point \(q\), the midpoint of line \(CB\):

\(q = \left( \displaystyle \frac{x_c + x_b}{2}, \, \displaystyle \frac{y_c + y_b}{2} \right)\)

Consider the slope of line \(pq\)

\(m\)_\(pq\)\(= \left( \displaystyle \frac{y_q - y_p}{x_q - x_p} \right)\)

\(= \left( \displaystyle \frac{ \left( \displaystyle \frac{y_c + y_b}{2} \right) - \left( \displaystyle \frac{y_a + y_b}{2} \right)}{ \left(\displaystyle \frac{x_c + x_b}{2} \right) - \left( \displaystyle \frac{x_a + x_b}{2} \right)} \right)\)

\(= \left( \displaystyle \frac{(y_c + y_b) - (y_a + y_b)}{(x_c + x_b) - (x_a + x_b)} \right) \)

\(= \displaystyle \frac{y_c - y_a}{x_c - x_a}\)

Thus, we have shown that \(m\)_\(AC\)\(\, = m\)_\(pq\). QED.

Shoelace formula:

Let \((x_1,y_1), (x_2,y_2), \cdots, (x_n,y_n)\) be coordinates of the vertices of a polygon with \(n\) sides. Then the area of the polygon given by \[A= \frac{1}{2} |x_1y_2+x_2y_3+\cdots+ x_ny_1-x_2y_1-x_3y_2-\cdots-x_1y_n|\]

Notice the blue laces and red laces in the attached figure.

Example \(\PageIndex{13}\):

Find the area of the triangle with vertices \((1,2),(2,3)\) and \((3,5)\). Make a shoelace diagram first!

By using the shoelace formula,

The area \(A= \frac{1}{2} |(1)(3)+(2)(5)+(3)(2)-(2)(2)-(3)(3)-(1)(5)|= \frac{1}{2}\).

 

Ruler and Compass Constructions

Compass: A tool for marking a circle.

Straightedge: A ruler without any markings.

Geometric constructions are delightful by nature; they bring playfulness to mathematics and offer a new form of creativity. But how do these constructions work, and what methodologies or proofs have emerged throughout history? Let’s uncover a few. To begin, we consider constructing a bisector of a given line.

Construction 1: Bisecting a line

Say we are given a line AB.

Our first step is to choose point A as the center point for our compass. Extend the compass to create a circle with a radius a little larger than half the length of AB, and draw the circle. From the other point B, draw a circle with the same radius, centered at B. The 2 points of intersection of the two circles will be points on
the straight line CD that bisects AB.

But how can we prove that the line \(CD\) is actually a bisector of the line  \(AB\)? One method uses congruent triangles to show geometrically that \(CD\) must bisect \(AB\).
If we connect the points as follows, we see that we’ve formed two isosceles triangles, \(DAC\) and \(DBC\). Because the circles created on points \(A \) and \(B \) have the same radius, we know that points \(D \) and \(C \) are an equal distance away from points \(A \) and \(B \); thus, \(DB = BC = CA = AD \). This demonstrates that the two triangles, \(DAC\) and \(DBC\), are isosceles.

Both triangles also share the same base, \(DC\). We know that congruent triangles must have the same angles and side lengths. Since triangles \(DAC \) and \(DBC \) share side \(DC \) and have two sides equal to \(DB \), they have three identical sides and are thus congruent. Congruent triangles have the
same height as each other; thus, we know that \(AE\) and \(EB \) are the same length, which proves that the line we’ve created is indeed a bisector of line \(AB \).

Construction 2: Creating a hexagon, given a line

Given a line FG, create a hexagon with a side length FG.

Using a compass, draw a circle with radius FG and center F. Create another circle with radius FG and center G. Extend line FG to create a new line FI of length 2(FG).
Construct yet another circle with radius FG and center point I. Mark each point of intersection among the three circles. Using points F, H, J, I, K, L as vertices, construct a hexagon.

A regular hexagon can be constructed from six congruent equilateral triangles. Using circles of equal radius enables the construction of multiple equilateral triangles that, in turn, form a hexagon. For example, we see that triangle FHG is equilateral. Point H is equidistant from F and G, and HF and HG are also equal to FG. Therefore, FHG must be equilateral. It follows that FLG maintains the same criteria and is an equilateral triangle congruent to FHG.

Furthermore, it follows that the pattern of construction around point I will also result in congruent equilateral triangles GJI and GKI, for circles of the same radius on I and G will form equilateral triangles much the same way that circles of the same radius formed equilateral triangles on F and G. Given that the angles FGH and IGJ sum to an angle of 120 degrees, it follows that the supplementary angle HGJ will be 60 degrees. We know that lines GH and GJ are equal and congruent to lines FG and IF thus HGJ is an equilateral triangle congruent to FHG. Nearly identical processing will construct triangle LGK, and we will see a hexagon with vertices F, H, J, I, K, L. Alternative methods for constructing a regular hexagon with side length FG might include bisecting angles, extending lines, creating congruent angles, and other techniques. As with many other problems, the potential solutions are infinite, provided the problem is actually solvable.

Activity

Construct a regular dodecagon using only a compass and a straightedge.

Hint

Start by drawing a circle with center O. Then pick any point on the circle, say A. Now, construct a circle with center A and the same radius as the previous one. Continue to construct seven circles using the intersecting points.

The following app/game may help in the construction:

https://www.euclidea.xyz/en/game/packs

For hundreds of years, three problems endlessly tormented mathematicians. Many mathematicians strove endlessly to find solutions to the problems, now
known as the “three classical problems.”

Theorem \(\PageIndex{1}\) Impossibility of Classical Compas and Striaghtedge Constructions

The three classical problems of antiquity listed below cannot be solved using only an unmarked straightedge and compass:

1. Squaring the Circle: Constructing a square with the same area as a given circle requires constructing the length \(\sqrt{\pi}\). Because \(\pi\) was proven to be a transcendental number (Lindemann, 1882), \(\sqrt{\pi}\) is not constructible.

2. Doubling the Cube: Constructing a cube with twice the volume of a unit cube requires constructing \(\sqrt[3]{2}\). Since the minimal polynomial of \(\sqrt[3]{2}\) over \(\mathbb{Q}\) is \(x^3 - 2 = 0\), its degree is \(3\), which is not a power of \(2\) (Wantzel, 1837).

 

3. Trisecting an Arbitrary Angle: Trisecting an arbitrary angle (such as \(60^\circ\), which requires constructing \(\cos(20^\circ)\)) involves solving a cubic equation \(8x^3 - 6x - 1 = 0\) over \(\mathbb{Q}\), whose degree is also \(3 \neq 2^k\).

Source:

By Job Bouwman [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)], from Wikimedia CommonsBy derivative work: Pbroks13 (talk) Square root of 2 triangle.png: en:User:Fredrik Simplified drawing: Rubber Duck (☮ • ✍) (Square root of 2 triangle.png) [Public domain], via Wikimedia Commons

By Pbroks13 at English Wikipedia [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

By Cmglee (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

By Vancho at English Wikipedia (Transferred from en.Wikipedia to Commons.) [Public domain], via Wikimedia Commons

By Cmglee (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

By http://commons.wikimedia.org/wiki/Us...r_Mathematicae (File:Secciones_cónicas.svg) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

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Thanks to Hannah Rayner

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