Triangle RST is drawn on the Coordinate plane with vertices at (5,-6), (5,0), and (7,2). The triangle is then translated 2 units to the right and 4 units down to create triangle R'S'T. what is the sun of the angles of triangle R'S'T? (1 point)
To find the sum of the angles of triangle R'S'T, we first need to find the measures of each angle in triangle RST.
To do this, we can use the distance formula to find the lengths of the sides of triangle RST:
Distance between (5,-6) and (5,0):
d = sqrt((5 - 5)^2 + (-6 - 0)^2)
d = sqrt(0 + 36)
d = sqrt(36)
d = 6
Distance between (5,0) and (7,2):
d = sqrt((7 - 5)^2 + (2 - 0)^2)
d = sqrt(4 + 4)
d = sqrt(8)
d = 2√2
Distance between (7,2) and (5,-6):
d = sqrt((5 - 7)^2 + (-6 - 2)^2)
d = sqrt(4 + 64)
d = sqrt(68)
d = 2√17
Since triangle RST is isosceles (the sides RT and ST have the same length), we can find the measure of angle R using the Law of Cosines:
RT^2 = d^2 + d^2 - 2d(d)(cos(R))
RT^2 = 36 + 36 - 2(6)(6)(cos(R))
RT^2 = 72 - 72(cos(R))
RT^2 = 72 - 72(cos(R))
0 = 72 - 72(cos(R))
0 = cos(R)
Since the cosine of any angle is never negative, cos(R) = 0 only when R = 90 degrees.
Therefore, the measure of angle R in triangle RST is 90 degrees.
Since triangle RST is an isosceles triangle, the measure of angles S and T are also 45 degrees each.
Now, since triangle R'S'T is a translation of triangle RST 2 units to the right and 4 units down, the position of each vertex will change accordingly.
The vertex R' will be (5 + 2, -6 - 4) = (7, -10).
The vertex S' will be (5 + 2, 0 - 4) = (7, -4).
The vertex T' will be (7 + 2, 2 - 4) = (9, -2).
Thus, triangle R'S'T will have vertices at (7, -10), (7, -4), and (9, -2).
To find the sum of the angles in triangle R'S'T, we can find the measures of each angle in the same manner as we did for triangle RST.
The measure of angle R' can be found using the Law of Cosines:
R'T'^2 = d^2 + d^2 - 2d(d)(cos(R'))
R'T'^2 = (2√2)^2 + (2√2)^2 - 2(2√2)(2√2)(cos(R'))
R'T'^2 = 8 + 8 - 8(cos(R'))
R'T'^2 = 16 - 16(cos(R'))
0 = 16 - 16(cos(R'))
R' = cos^(-1)(0)
R' = 90 degrees
Similarly, the measure of angle T' is also 90 degrees.
Since triangle R'S'T is an isosceles triangle, the measure of angle S' is also 90 degrees.
Therefore, the sum of the angles in triangle R'S'T is 90 + 90 + 90 = 270 degrees.