Answer:
The Area of Δ TOP is 43.55 units².
Step-by-step explanation:
Given:
P ≡ ( x₁ ,y₁ ) ≡ ( -5 , -7)
T ≡ ( x₂ ,y₂ ) ≡ ( 1 , 8)
O ≡ ( x₃ ,y₃ ) ≡ ( 6 , 6)
To Find :
Area of Δ TOP = ?
Solution :
We have
![\textrm{Area of Triangle TOP} = \frac{1}{2}\times Base\times Height\\\textrm{Area of Triangle TOP} = \frac{1}{2}\times OT\times PT](https://tex.z-dn.net/?f=%5Ctextrm%7BArea%20of%20Triangle%20TOP%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20Base%5Ctimes%20Height%5C%5C%5Ctextrm%7BArea%20of%20Triangle%20TOP%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20OT%5Ctimes%20PT)
Now Distance formula we have
![l(PT) = \sqrt{((x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} )}](https://tex.z-dn.net/?f=l%28PT%29%20%3D%20%5Csqrt%7B%28%28x_%7B2%7D-x_%7B1%7D%29%5E%7B2%7D%2B%28y_%7B2%7D-y_%7B1%7D%29%5E%7B2%7D%20%29%7D)
![l(OT) = \sqrt{((x_{3}-x_{2})^{2}+(y_{3}-y_{2})^{2} )}](https://tex.z-dn.net/?f=l%28OT%29%20%3D%20%5Csqrt%7B%28%28x_%7B3%7D-x_%7B2%7D%29%5E%7B2%7D%2B%28y_%7B3%7D-y_%7B2%7D%29%5E%7B2%7D%20%29%7D)
Substituting the given values we get
![l(PT) = \sqrt{((1--5)^{2}+(8--7)^{2} )}\\l(PT) = \sqrt{((1+5)^{2}+(8+7)^{2} )}\\l(PT) = \sqrt{((6)^{2}+(15)^{2} )}\\l(PT) = \sqrt{261}\\l(PT) = 16.16\ units](https://tex.z-dn.net/?f=l%28PT%29%20%3D%20%5Csqrt%7B%28%281--5%29%5E%7B2%7D%2B%288--7%29%5E%7B2%7D%20%29%7D%5C%5Cl%28PT%29%20%3D%20%5Csqrt%7B%28%281%2B5%29%5E%7B2%7D%2B%288%2B7%29%5E%7B2%7D%20%29%7D%5C%5Cl%28PT%29%20%3D%20%5Csqrt%7B%28%286%29%5E%7B2%7D%2B%2815%29%5E%7B2%7D%20%29%7D%5C%5Cl%28PT%29%20%3D%20%5Csqrt%7B261%7D%5C%5Cl%28PT%29%20%3D%2016.16%5C%20units)
And
![l(OT) = \sqrt{((x_{3}-x_{2})^{2}+(y_{3}-y_{2})^{2} )}\\l(OT) = \sqrt{((6-1)^{2}+(6-8)^{2} )}\\l(OT) = \sqrt{((5)^{2}+(-2)^{2} )}\\l(OT) = \sqrt{29}\\l(OT) = 5.39\ units](https://tex.z-dn.net/?f=l%28OT%29%20%3D%20%5Csqrt%7B%28%28x_%7B3%7D-x_%7B2%7D%29%5E%7B2%7D%2B%28y_%7B3%7D-y_%7B2%7D%29%5E%7B2%7D%20%29%7D%5C%5Cl%28OT%29%20%3D%20%5Csqrt%7B%28%286-1%29%5E%7B2%7D%2B%286-8%29%5E%7B2%7D%20%29%7D%5C%5Cl%28OT%29%20%3D%20%5Csqrt%7B%28%285%29%5E%7B2%7D%2B%28-2%29%5E%7B2%7D%20%29%7D%5C%5Cl%28OT%29%20%3D%20%5Csqrt%7B29%7D%5C%5Cl%28OT%29%20%3D%205.39%5C%20units)
Now substituting OT and PT in area formula we get
![\textrm{Area of Triangle TOP} = \frac{1}{2}\times 5.39\times 16.16\\\textrm{Area of Triangle TOP} = 43.55\ units^{2}](https://tex.z-dn.net/?f=%5Ctextrm%7BArea%20of%20Triangle%20TOP%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Ctimes%205.39%5Ctimes%2016.16%5C%5C%5Ctextrm%7BArea%20of%20Triangle%20TOP%7D%20%3D%2043.55%5C%20units%5E%7B2%7D)
Therefore, Area of Δ TOP is 43.55 units².