Given : A (8,5), B (7,2), C (4,6)A(8,5),B(7,2),C(4,6), h = 7#
Using distance formula we can calculate the lengths of sides a, b, c.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)d=√(x2−x1)2+(y2−y1)2
a = sqrt((7-4)^2+(2-6)^2) = 5a=√(7−4)2+(2−6)2=5
b = sqrt((4-8)^2+(6-5)^2) = sqrt 17 b=√(4−8)2+(6−5)2=√17
c = sqrt((8-7)^2+(5-2)^2) = sqrt 10c=√(8−7)2+(5−2)2=√10
Semi perimeter of base triangle s = (a+b+c)/2s=a+b+c2
s = (5 + sqrt 17 + sqrt 10)/2 = 12.2854/2 ~~ 6.2427s=5+√17+√102=12.28542≈6.2427
Area of triangle A_t = sqrt(s (s-a) (s-b) (s-c))At=√s(s−a)(s−b)(s−c)
A_t = sqrt(6.2427 * (6.2427-5)*(6.2427- sqrt 17)*(6.2427-sqrt 10))At=√6.2427⋅(6.2427−5)⋅(6.2427−√17)⋅(6.2427−√10)
#A_t ~~ 7.1171
Volume of pyramid V = (1/3) * A_t * hV=(13)⋅At⋅h
color(green)(V = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"V=(13)⋅7.1171⋅7=16.6065, cubic units