High School Geometry

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Last Updated: Sunday, 15 September 2024 22:27
Published: Tuesday, 31 January 2023 00:30
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High School Geometry:

We learned simple geometry in Elementary maths. More advanced topics for geometry and trigonometry can be found at the bottom of the page on this link: https://www.mathsisfun.com/geometry/index.html

 

Coordinate geometry: This refers to drawing 2D shapes on a coordinate axis (horizontal X-axis and vertical Y axis). To draw 3D shpaes, we'll need 3 axis (X, Y and Z axis) which is more complex. We'll focus on 2D geometry for now, and explore simple 3D shapes.

Distance between 2 points (x1,y1) and (x2,y2) = √[(x2-x1)^2 + (y2-y1)^2] (This can be done by using Pythagoras Thm)

Distance between a point (x1,y1) and a line (ax+by+c=0) => Draw a perpendicular from the point to the line, and find the eqn of that line (since slope and one of the points is known). Now find intersecting point for the 2 lines. Calc distance b/w the 2 points, which is the distance asked. Easier than this is to use a formula prrof of which is here: https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

2D shapes: Polygons are 2-dimensional shapes made of straight lines. These include triangles, rectangles, etc. Here we calculate area and perimeter. attach diagram FIXME

  • Line/Point. These are not shapes. A line is made up of inifinite points. Lines are straight or curved. We study straight lines since they are easy.
  • Square/rectangle:
  • Triangle:
  • Parallelogram:
  • Rhombus:
  • Kite:
  • Polygons: (

3D shapes: Studied in Solid Geometry. It's called "solid", as we can make solid shapes only from 3D objects. Nothing we see around us is 2D. Everything is 3D. Here we usually calculate Volume and surface area of these objects. There are two main types of solids, "Polyhedra", and "Non-Polyhedra"

  • Polyhedra: All surfaces are flat.
  • Non Polyhedra: At least one of the surfaces is NOT flat.

 

Bisect Lines/Angles:

To bisect a line into 2 halves, we use  technique with compass, where we draw arcs on top and bottom of the lines from the 2 end poibts. Wherever they intersect, we draw a straight line thru it, that line divides the original line into 2 halves. Why does it work? Because, we make a rhombus and the 2 lines become the diagonal of rhombus.

 

Circles:

We learned about Circles in "Elementary Maths", but there are a lot of properties of circles and triangles inside or outside the circles that yields a lot of interesting theorems.

  • Incenter: A circle which touches the 3 sides of a triangles is completely inside the triangle and is called the incircle. The centre of incircle is called the incenter and can be found by dividing angles of each vertex in half and finding the intersecting point of these 3 angle bisectors. 2 adjacent triangles turn out to be similar, which proves that such a point is incenter of the triangle. There's also 3 excircles defined which touch the 3 exterior or extended sides of the triangle. Excircles are not usually discussed in high school geometry.
  • Circumcenter: A circle which passes thru the 3 vertices of a triangle is called the Circumcircle, and the center of such a circle is called Circumcenter. Circumcenter can be found by drawing perpendicular bisector of the 3 sides. The intersecting point is the circumcenter. It can be proved that such a point is equidistant from the 3 vertices by observing that the 2 triangles on each side of triangle are congruent.
  • Orthocenter: Orthocenter of a triangle is the point where the 3 altitudes of the triangle coincide. The perpendicular is drawn by drawing altitude from each of the 3 vertices on to the opposite side. Orthocenter doesn't have any special property as Incenter or circumcenter have.
    • Link: https://en.wikipedia.org/wiki/Altitude_(triangle)#Orthocenter
    • One important property of orthocentre for a right trangle is: square altitude (h) from right angle to hypotenuse = product of the lengths of hypotenuse segments divided by the altitude, i.e h^2 = √(p*q)
    • The orthocenter lies inside the triangle iff the triangle is acute. If one angle is a right angle, the orthocenter coincides with the vertex at the right angle.
    • 3 altitude of the triangle can be found by using heron's formula below (by using 1/2*base*height=Area)
    • Both inradius and circumradius of the traingle are related to the height of the 3 sides via inradius and circumradius theorem. See wikilink above for the relation.
    • Orthic or Altitude triangle: The feet of the altitudes of an oblique triangle form the orthic triangle. Also, the incenter (the center of the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.
  • Centroid: Centroid also known as geometric center or center of figure, of a figure is the arithmetic mean position of all the points in the surface of the figure. For a object with uniform mass, it's also the center of gravity. The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). Mathematicaly, centroid is the mean of the 3 coordiantes, i.e coordinates of centroid (x,y) = ((x1+x2+x3)/3, ((y1+y2+y3)/3) where (x1,y1), (x2,y2) and (x3,y3) are the 3 coordiantes.

 

 

Trignometry (Triangle Geometry):

Trignometry is a branch of Geometry that deals exclusively with triangles. You may wonder how come triangles have a whole branch of Mathematics dedicated to itself !! There are lots of things that's possible with triangles, and knowing triangles well forms the basis of Geometry.We'll talk more about right angled triangles in separate section on "Trignometry",as they are the most interesting ones. Here we just go thru few basics.

 

 

Finding Sides and angles of any Generic triangle:

So far, we looked at right angle triangles. We figured out that given 2 sides of a right angled triangle, we can find out 3rd side by using Pythagoras theorem. Then we can find out the 2 angles of the triangle by using trignometric tables (by finding out sin,cos or tan ratio and then seeing what angle corresponds to that ratio). We also figured out that given 1 side and 1 angle, we can again create a unique triangle. Here we can find remaining sides by using trignometric tables. So, right angles are easy.

How about finding sides and angles of a unique triangle which is not right triangle? It's possible to do it via 2 laws (both laws are actually same law, but written differently. One can be derived from the other. Below are the 2 laws:

1. Law of Sines:

This is an important theorem to find length of sides or angles of a triangle given it's 2 sides and an angle or an angle and 2 sides (i.e given some combination of length of sides and angles that makes it unique). It is used to find the radius of circumcenter too. Stated mathematically, this is the law of Sines:

Sin(A)/a = Sin(B)/b = Sin(C)/c =1/2R where a,b.c are the lengths of 3 sides, and A,B,C are 3 angles opposite the 3 sides (i.e angle A is angle opposite side a, meaning angle between sides b and c), R is the circumcenter.

In other words, a:b:c = Sin(A):Sin(B):Sin(C) => i.e sides are proportional to the sine of the respective angles.

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_sines

Proof: Proof is very simple. If you find area of any triangle by multiplying base and height in 3 different ways, then we can get this equality. There is one more proof based on inscribing a triangle around a circle. That proof is also given in the wikipedia link above.

2. Law of Cosines:

This is a variation of the Law of Cosines. Here given 2 sides and the angle between them, we can find the 3rd side much easily. Law of Sines won't give us the 3rd side that easily. Law of Cosines is a more generic case of Pythagoras theorem, where it applies to angles other than 90 degrees. Stated mathematically, this is the law of Cosines:

c^2 = a^2 + b^2 -2abCos(C) where a,b.c are the lengths of 3 sides, and C is angle between sides a and b, i.e opposite side c)

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_cosines

Proof: Proof is remarkably simple if we use coordinate system, with one vertex of triangle placed at (0,0). We make a right triangle out of the given triangle, with the extra 2 legths of the right angle triangle being bCos(C) and bSin(C). Now we apply Pythagoras thm, so that c^2=(a+bCos(C))^2 + (bSin(C))^2. Rearranging terms and using identity sin2(Θ) + cos2(Θ) = 1, we get the formula above.

NOTE: There is -ve sign on the last term. A very simple way to remember this is as follows =>If angle C was right angle, then we get back to Pythagoras thm via Law of Cosines (by using Cos(C)=0). If angle C is acute angle, then "c" would be smaller than the one for a right angle triangle, so c^2 has to be less than a^2+b^2. So, we need to subtract some term from this, which is done by having a -ve sign to 2ab. Cos(C) is +ve for acute angle. For obtuse angle, Cos(C) is -ve, which makes the 3rd term +ve (-2ab*(-ve value) = +ve value). So, c^2 becomes greater than a^2+b^2 which is what is expected.

3. Law of Cotangent (or Cot Theorem):

Though not so common, Law of Cot provides a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. It is used to find the radius of inscribed circle too. It states as follows:

Cot(A/2)/(s-a) = Cot(B/2)/(s-b) = Cot(C/2)/(s-c) = 1/r where s is the semiperimeter of the triangle i.e s = 1/2(a+b+c), and r is the radius of the inscribed circle.

Furthermore inradius is also given by (only interms of sides and no angles) => r = sq root ((s-a)(s-b)(s-c)/s)

More info is given on wikipedia here: https://en.wikipedia.org/wiki/Law_of_cotangents

Proof: Proof is on wiki link above.

Heron's Formula:

Using the law of cosines, or by using pythagoras theorem, we can find out the area of a triangle given it's 3 sides. Law of Cot can also be used to derive this. This is known as Heron's formula as stated below.

Given 3 sides of a triangle as a,b,c, it's Area = √(s(s-a)(s-b)(s-c)) where s is the semiperimeter of the triangle i.e s = 1/2(a+b+c)

Wikipedia link proves it here: https://en.wikipedia.org/wiki/Heron's_formula

Heron's formula is special case of Brahmagupta formula, which is a special case of Bretschneider's formula for finding out area of any quadrilateral: https://en.wikipedia.org/wiki/Bretschneider's_formula

Triangle Theorems:

  1. Menelaus's Thm: It relates the ratios obtained by cutting 3 sides of a triangle. Wiki: https://en.wikipedia.org/wiki/Menelaus%27s_theorem

Circle Theorems:

  1. Thales's Thm and others related to triangles inside circles: Link => https://www.mathsisfun.com/geometry/circle-theorems.html