Explanation:
The two adjacent vertices of the triangle are A(5; 3) and B(7; 7), and one point on the opposite side is P(-2; 9). To find the coordinates of the third vertex, we need to find the midpoint of AB and extend it to P.
The midpoint of AB can be calculated by finding the average of their x-coordinates and y-coordinates:
C = ((A_x + B_x)/2, (A_y + B_y)/2) = ((5 + 7)/2, (3 + 7)/2) = (6, 5)
Next, we can use the midpoint C and point P to find the slope of the line connecting them:
m = (P_y - C_y) / (P_x - C_x) = (9 - 5) / (-2 - 6) = 4 / -8 = -0.5
Since the slope of a line perpendicular to this one is -2, we can use the midpoint C and the slope to find the equation of the line connecting C and the third vertex D:
y - C_y = -2 (x - C_x)
y - 5 = -2 (x - 6)
Expanding and solving for x, we get:
y = -2x + 17
Substituting the y-coordinate of P (-2) into this equation:
-2 = -2x + 17
Solving for x:
x = 7.5
Finally, we can use the x-coordinate to find the y-coordinate:
y = -2x + 17 = -2 * 7.5 + 17 = 5.5
So the third vertex is D (7.5, 5.5).
The vertices of the triangle are A (5, 3), B (7, 7), and D (7.5, 5.5).