- #1
Albert1
- 1,221
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In summary, the formula for finding the area of a triangle is A = 1/2 * base * height, and the units for area are typically square units. The area of a triangle cannot be negative, and if the base and height are not given, other measurements can be used to find the area. The area of a triangle is directly proportional to the lengths of its sides, meaning that changes in side lengths will result in corresponding changes in the area.
- #2
Amer
- 259
- 0
First I will find D coordinates D(x,y) .
the slope of OD: $\displaystyle S_{OD} = \frac{y}{x} $
Note that $\displaystyle y = \frac{-3x}{4} + 2 $ since D on the line
$\displaystyle AD \perp OD $ so we get $S_{AD} . S_{OD} = -1 $
$\displaystyle S_{AD} = \frac{-3}{4} $
$\displaystyle S_{OD} = \frac{4}{3} = \frac{-3x + 8}{4x} \Rightarrow 16x = -9x + 24 \Rightarrow x = \frac{24}{25} $
$D\left(\frac{24}{25} , \frac{32}{25} \right) $
$\displaystyle \bar{OD} = \sqrt{\left( \frac{24}{25}\right)^2 + \left( \frac{32}{25}\right)^2} = \frac{8}{5} $
The upper circle has radius 1 since the line AB cut the y-axis at (0,2), I think the lower circle has also radius 1, if so then N(1,1)
to find G coordinates, G on the line ON : y=x so $\displaystyle G\left(\frac{8}{7}, \frac{8}{7} \right) $
$\displaystyle\bar{DG} = \sqrt{ \left(\frac{24}{25} - \frac{8}{7}\right)^2 + \left(\frac{32}{25} - \frac{8}{7}\right)^2 }= 8 \sqrt{\left(\frac{3}{25} - \frac{1}{7}\right)^2 + \left(\frac{4}{25} - \frac{1}{7} \right)^2 } $
The area of the triangle = 1/2 OD (DG)
the slope of OD: $\displaystyle S_{OD} = \frac{y}{x} $
Note that $\displaystyle y = \frac{-3x}{4} + 2 $ since D on the line
$\displaystyle AD \perp OD $ so we get $S_{AD} . S_{OD} = -1 $
$\displaystyle S_{AD} = \frac{-3}{4} $
$\displaystyle S_{OD} = \frac{4}{3} = \frac{-3x + 8}{4x} \Rightarrow 16x = -9x + 24 \Rightarrow x = \frac{24}{25} $
$D\left(\frac{24}{25} , \frac{32}{25} \right) $
$\displaystyle \bar{OD} = \sqrt{\left( \frac{24}{25}\right)^2 + \left( \frac{32}{25}\right)^2} = \frac{8}{5} $
The upper circle has radius 1 since the line AB cut the y-axis at (0,2), I think the lower circle has also radius 1, if so then N(1,1)
to find G coordinates, G on the line ON : y=x so $\displaystyle G\left(\frac{8}{7}, \frac{8}{7} \right) $
$\displaystyle\bar{DG} = \sqrt{ \left(\frac{24}{25} - \frac{8}{7}\right)^2 + \left(\frac{32}{25} - \frac{8}{7}\right)^2 }= 8 \sqrt{\left(\frac{3}{25} - \frac{1}{7}\right)^2 + \left(\frac{4}{25} - \frac{1}{7} \right)^2 } $
The area of the triangle = 1/2 OD (DG)
- #3
Albert1
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Albert said:
Attachments
Related to What is the Area of Triangle ODG?
1. What is the formula for finding the area of a triangle?
The formula for finding the area of a triangle is A = 1/2 * base * height, where A represents the area, base represents the length of the base of the triangle, and height represents the height of the triangle.
2. What are the units for area of a triangle?
The units for area of a triangle are typically square units, such as square inches, square feet, or square meters.
3. Can the area of a triangle be negative?
No, the area of a triangle cannot be negative. It is always a positive value, as it represents the amount of space enclosed within the triangle.
4. How do you find the area of a triangle if the base and height are not given?
If the base and height of a triangle are not given, you can use other measurements such as the length of the sides or the angles to find the area. There are various formulas and methods for doing this, such as Heron's formula or trigonometric functions.
5. What is the relationship between the area of a triangle and its sides?
The area of a triangle is directly proportional to the lengths of its sides. This means that if you increase the length of one or more sides, the area will also increase. Similarly, if you decrease the length of one or more sides, the area will decrease.
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