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Mathematics LibreTexts

4.2: 2-D Geometry

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    4888
  • [ "article:topic", "closed curve", "polygon", "Convex polygons", "Concave polygons", "apothem", "authorname:thangarajahp" ]

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    Two-dimensional figures

    Definition

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

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

    A curve is called closed curve if we can trace the figure in such a way that our starting point and ending point are the same.

    A simple closed curve is a curve that we can trace without going over any point more than once while starting point and ending point are the same.

    Polygons

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

    Example \(\PageIndex{1}\):

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

    Definitions

    Convex polygons are polygons which, if a line is drawn across them at any point, only two edges will be intersected by the line.

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

    Convex and Concave Polygons

    A polygon is considered to be regular if all of its sides have the same measure. By definition, this also implies that all of the polygon's internal angles will be the same.

    diagonal is a line which 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 which 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 centre to the edge.

    Regular Polygons

    Properties

    A regular polygon's apothem is a line which is drawn from its center to the midpoint of one of its faces

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

    The perimeter of a shape is the sum of the length of all its faces

    The area of a polygon is the measure of its surface area, 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 with all sides 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.

     

    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 that connects a vertex to the side opposite to that vertex.

    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 equivalent.

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

    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, we can say that their areas are equivalent.

    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 we removed the same quantity of area from each.

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

    Method 2: President Garfield method

     

    By Cflm001 [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

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

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

     

    A Kepler Triangle is a right angle triangle with sides \( 1, \phi, \phi^2), where \( \phi\) is the golden ratio.

    Example \(\PageIndex{7}\):

     

    Triangle inequality

    By WhiteTimberwolf, Brews ohare (PNG version) (PNG version) [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

    \(x+y \geq z\).

    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

    Triangles are said to be similar triangles if their internal angles are the same, but their sides are different sizes. The sides or 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 both their sides and their angles are the same.

    Quadrilaterals

    Quadrilaterals are polygons with four edges and four vertices. 

    • A kite is a quadrilateral with adjacent pairs of sides are equal.
    • A trapezoid is a quadrilateral with one pair of parallel sides.
    • A parallelogram is a quadrilateral with opposite sides that are parallel and equal.
    • A rhombus is a parallelogram that has all four sides equal.
    • A rectangle has four right angles, and thus the sides are parallel and equal in pairs (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,  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 which 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 make this shape into an aggregate of shapes we know more about: triangles.

    As we can see, a quadrilateral is made of two triangles. Since we know that the sum of the internal angles of each triangle is \(180^\circ\), we can add those two together 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 between a rhombus, square, and rectangle with a Venn diagram.

    Pentagrams

    Pentagrams are polygons that make a five-pointed star.

    Example \(\PageIndex{9}\): Proof: The sum of Vertex Angle of a Pentagram is 180°

    Consider the following pentagram:

    As we can see, we have determined the interior angles of the black triangle in the general case, using the internal vertex angles from 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 are 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 they have an equal number of sides which are proportional to each other in the same ratio.

    Example \(\PageIndex{8}\):

    Tessellation

    Definition

    Tessellation refers to the "tiling" of polygons, as in ceramic tiles or quilts. There are regular tessellation and semi-regular tessellation. 

    ,@api,deki,files,7329,quilt1.pdf

     

    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?

     

     

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

    Picture2.jpg

     

    Curved figures

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

    Circle Geometry

    Definitions

    The center of a circle is a point in the circle which is equidistant from all points along the circle's edge. This distance is given by the radius.

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

    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 an angle given by two chords that share a common endpoint.

    central angle is an angle describing an arc. This is measured, in circles, in radians.

    Properties
    1. The perpendicular line from the center of a circle to a chord bisects the chord.
    2. \(\pi=\frac{ Circumference}{Diameter}\).
    3. \(A = \pi r\)\(2\)
    4. The angle between the tangent at a point, and the radius from 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. To find distances between points, we create the right triangles and apply the Pythagorean Theorem.

    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.

    Midpoint Coordinates:

    The midpoint coordinates of a line can be thought of as the average of both \(A\) and \(B\) coordinates from both endpoints. So, the formula for the midpoint of a line is:

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

    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 indicating 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 \(\PageIndex{12}\):

    Show, by using coordinate geometry, 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 shoe lace 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 marks on it.

    The following app/game may help in the constructions:

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

    Activity

    Construct a regular dodecagon using only a compass and straight edge.

    Hint

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

     

    Example \(\PageIndex{14}\):

    Using an unmarked straight edge and a compass, try to do the following exercises:

    • Square a circle - given a circle of a certain area, create a square with the same area.

    • Trisect an Angle - divide an angle into three equal parts.

    • Construct a cube with twice the volume of a given cube.

     

    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 

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