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Grade 10/Math/MPM2DZ/Math Oral Presentation Questions/Analytical Geometry.md
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Lets first find each of the side lengths to determine if the triangle is **obtuse**, **acute** or scalene.
$`AB = \sqrt{(-1-7)^2 + (5-2)^2} = \sqrt{64 + 9} = \sqrt{73}`$
$`\overline{AB} = \sqrt{(-1-7)^2 + (5-2)^2} = \sqrt{64 + 9} = \sqrt{73}`$
$`BC = \sqrt{(7-(-1))^2 + (2-(-4))^2} = \sqrt{64 + 36} = \sqrt{100} = 10`$
$`\overline{BC} = \sqrt{(7-(-1))^2 + (2-(-4))^2} = \sqrt{64 + 36} = \sqrt{100} = 10`$
$`AC = \sqrt{(-1-(-1))^2 + (5-(-4))^2} = \sqrt{0^2 + 9^2} = \sqrt{81} = 9`$
$`\overline{AC} = \sqrt{(-1-(-1))^2 + (5-(-4))^2} = \sqrt{0^2 + 9^2} = \sqrt{81} = 9`$
$`\since
$`\because \overline{AB} =\not \overline{BC} =\not \overline{AC}`$
$`\therefore \triangle ABC`$ is a scalene triangle.
### Question 1 b)
The `orthocenter` is the POI of the heights of a triangle.
$`m_{AB} = \dfrac{2-5}{7-(-1)} = \dfrac{-3}{8}`$
$`m_{\perp AB} = \dfrac{8}{3}`$
$`y_{\perp AB} - (-4) = \dfrac{8}{3}(x - (-1)) \implies y_{perp AB} + 4 = \dfrac{8}{3}(x+1)`$
$`y_{\perp AB} = \dfrac{8}{3}x + \dfrac{8}{3} - 4`$
$` y_{\perp AB} = \dfrac{8}{3}x - \dfrac{4}{3} \quad (1)`$
$`m_{BC} = \dfrac{2-(-4)}{7-(-1)} = \dfrac{6}{8} = \dfrac{3}{4}`$
$`m_{\perp BC} = \dfrac{-4}{3}`$
$`y_{\perp BC} - 5 = \dfrac{-4}{3}(x-(-1)) \implies y_{\perp BC} - 5 = \dfrac{-4}{3}(x+1)`$
$`y_{\perp BC} = \dfrac{-4}{3}x - \dfrac{4}{3} + 5`$
$`y_{\perp BC} = \dfrac{-4}{3}x + \dfrac{11}{3}`$
```math
\begin{cases}
y_{\perp AB} = \dfrac{8}{3}x - \dfrac{4}{3} & \text{(1)} \\
\\
y_{\perp BC} = \dfrac{-4}{3}x + \dfrac{11}{3} & \text{(2)} \\
\end{cases}
```
Sub $`(1)`$ into $`(2)`$:
$`\dfrac{8}{3}x - \dfrac{4}{3} = \dfrac{-4}{3} + \dfrac{11}{3}`$
$`8x - 4 = -4x + 11`$
$`12x = 15`$
$`x = \dfrac{5}{4} \quad (3)`$
Sub $`(3)`$ into $`(2)`$
$`y = \dfrac{-20}{12} + \dfrac{11}{3}`$
$`y = \dfrac{-5}{3} + \dfrac{11}{3}`$
$`y = \dfrac{6}{3} = 2`$
$`y = 2`$
$`\therefore`$ The `orthocenter` is at $`(\dfrac{5}{4}, 2)`$
### Question 2 a)
midpoint = $`\large (\dfrac{\sqrt{72} + \sqrt{32}}{2}, \dfrac{-\sqrt{12} - \sqrt{48}}{2} \large )`$