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Update Unit 1: Analytical Geometry.md

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James Su 2019-09-18 14:57:51 +00:00
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Grade 10/Math/MPM2DZ/Unit 1: Analytical Geometry.md
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@ -61,3 +61,41 @@ Let $`x_c, y_c`$ be the center
$`(x - x_c)^2 + (y - y_c)^2 = r^2`$
To get the center, just find a $`x, y`$ such that $`x - x_c = 0`$ and $`y - y_c = 0`$
## Triangle Centers
## Centroid
The centroid of a triangle is the common intersection of the 3 medians. The centroid is also known as the centre of mass or centre of gravity of an object (where the mass of an object is concentrated).
### Procedure To Determine The Centroid
1. Find the equation of the two median lines. **The median is the line segment from a vertex from a vertex to the midpoint of the opposite side**.
2. Find the point of intersection using elimnation or substitution.
- Alternatively, only for checking your work, let the centroid be the point $`(x, y)`$, and the 3 other points be $`(x_1, y_1), (x_2, y_2), (x_3, y_3)`$ respectively, then the
centroid is simply at $`(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1+y_2+y_3}{3})`$
## Circumcentre
The circumcentre ($`O`$) of a triangle is the common intersection of the 3 perpendicular bisectors of the sides of a triangle.
### Procedure To Determine The Centroid
1. Find the equation of the perpendicular bisectors of two sides. **A perpendicular (right) bisector is perpendicular to a side of the triangle and passes through the midpoint of that side of the triangle**.
2. Find the point of intersection of the two lines using elimination or substitution.
## Orthocentre
The orthocenter of a triangle is the common intersection of the 3 lines containing the altitudes.
### Procedure To Determine The Orthocentre
1.Find the equation of two of the altitude lines. **An altitude is a perpendicular line segment from a vertex to the line of the opposite side.**
2. Find the point of intersection of the two lines using elimination or substitution.
## Classifying Shapes
<img src="https://files.catbox.moe/3cfs4h.png" width="600">
## Properties Of Quadrilaterals
<img src="https://files.catbox.moe/asixh9.png" width="500">