Answer on Question #50570 – Math – Geometry
Graph (0,8) (6,5) (-3,2) and find the area.
Solution.
S = p ( p − a ) ( p − b ) ( p − c ) ; S = \sqrt {p (p - a) (p - b) (p - c)}; S = p ( p − a ) ( p − b ) ( p − c ) ; p = a + b + c 2 ; p = \frac {a + b + c}{2}; p = 2 a + b + c ; a = ( 6 − 0 ) 2 + ( 5 − 8 ) 2 = 45 ; a = \sqrt {(6 - 0) ^ {2} + (5 - 8) ^ {2}} = \sqrt {4 5}; a = ( 6 − 0 ) 2 + ( 5 − 8 ) 2 = 45 ; b = ( − 3 − 6 ) 2 + ( 2 − 5 ) 2 = 90 ; b = \sqrt {(- 3 - 6) ^ {2} + (2 - 5) ^ {2}} = \sqrt {9 0}; b = ( − 3 − 6 ) 2 + ( 2 − 5 ) 2 = 90 ; c = ( 0 − ( − 3 ) ) 2 + ( 8 − 2 ) 2 = 45 ; c = \sqrt {(0 - (- 3)) ^ {2} + (8 - 2) ^ {2}} = \sqrt {4 5}; c = ( 0 − ( − 3 ) ) 2 + ( 8 − 2 ) 2 = 45 ; p = 45 + 90 + 45 2 = 2 45 + 90 2 ; p = \frac {\sqrt {4 5} + \sqrt {9 0} + \sqrt {4 5}}{2} = \frac {2 \sqrt {4 5} + \sqrt {9 0}}{2}; p = 2 45 + 90 + 45 = 2 2 45 + 90 ; S = 2 45 + 90 2 ( 2 45 + 90 2 − 45 ) ( 2 45 + 90 2 − 90 ) ( 2 45 + 90 2 − 45 ) = S = \sqrt {\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} \left(\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} - \sqrt {4 5}\right) \left(\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} - \sqrt {9 0}\right) \left(\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} - \sqrt {4 5}\right)} = S = 2 2 45 + 90 ( 2 2 45 + 90 − 45 ) ( 2 2 45 + 90 − 90 ) ( 2 2 45 + 90 − 45 ) = 2 45 + 90 2 ⋅ 2 45 + 90 − 2 45 2 ⋅ 2 45 + 90 − 2 90 2 ⋅ 2 45 + 90 − 2 45 2 = \sqrt {\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} \cdot \frac {2 \sqrt {4 5} + \sqrt {9 0} - 2 \sqrt {4 5}}{2} \cdot \frac {2 \sqrt {4 5} + \sqrt {9 0} - 2 \sqrt {9 0}}{2} \cdot \frac {2 \sqrt {4 5} + \sqrt {9 0} - 2 \sqrt {4 5}}{2}} = 2 2 45 + 90 ⋅ 2 2 45 + 90 − 2 45 ⋅ 2 2 45 + 90 − 2 90 ⋅ 2 2 45 + 90 − 2 45 = 2 45 + 90 2 ⋅ 90 2 ⋅ 2 45 − 90 2 ⋅ 90 2 = ( 90 ) 2 2 2 ⋅ ( 2 45 ) 2 − ( 90 ) 2 2 2 = 90 4 ⋅ 180 − 90 4 = 90 4 = 22.5. \sqrt {\frac {2 \sqrt {4 5} + \sqrt {9 0}}{2} \cdot \frac {\sqrt {9 0}}{2} \cdot \frac {2 \sqrt {4 5} - \sqrt {9 0}}{2} \cdot \frac {\sqrt {9 0}}{2}} = \sqrt {\frac {(\sqrt {9 0}) ^ {2}}{2 ^ {2}} \cdot \frac {(2 \sqrt {4 5}) ^ {2} - (\sqrt {9 0}) ^ {2}}{2 ^ {2}}} = \sqrt {\frac {9 0}{4} \cdot \frac {1 8 0 - 9 0}{4}} = \frac {9 0}{4} = 2 2. 5. 2 2 45 + 90 ⋅ 2 90 ⋅ 2 2 45 − 90 ⋅ 2 90 = 2 2 ( 90 ) 2 ⋅ 2 2 ( 2 45 ) 2 − ( 90 ) 2 = 4 90 ⋅ 4 180 − 90 = 4 90 = 22.5.
Answer: 22.5.
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#339914 on Dec 2023