ABC and RST are similar and have the same orientation. The hypotenuse of RST is on the same line as the hypotenuse of ABC and is twice the length of AB.
What is the equation of the line that contains the hypotenuse of RST?
Since ABC and RST are similar and have the same orientation, their corresponding sides are proportional. The hypotenuse of RST is twice the length of AB. Therefore, we can express the relationship between the lengths of the corresponding sides as:
RT = 2(AB)
Now, we need to find the equation of the line that contains the hypotenuse of RST.
Let's assume that the coordinates of the endpoints of the hypotenuse of RST are (x1, y1) and (x2, y2).
Using the distance formula, we can find the length of the hypotenuse of RST:
RT = √((x2 - x1)^2 + (y2 - y1)^2)
Since RT is twice the length of AB, we can substitute 2(AB) for RT:
2(AB) = √((x2 - x1)^2 + (y2 - y1)^2)
Squaring both sides to eliminate the square root:
4(AB)^2 = (x2 - x1)^2 + (y2 - y1)^2
Since AB is the length of the hypotenuse of ABC, we can express it as:
(AB)^2 = (x2 - x1)^2 + (y2 - y1)^2
Simplifying the equation:
4[(x2 - x1)^2 + (y2 - y1)^2] = (x2 - x1)^2 + (y2 - y1)^2
Distributing the 4:
4x2 - 4x1)^2 + 4(y2 - y1)^2 = x2^2 - 2x2x1 + x1^2 + y2^2 - 2y2y1 + y1^2
Simplifying further:
4x2 - 4x1)^2 + 4(y2 - y1)^2 = x2^2 - 2x2x1 + x1^2 + y2^2 - 2y2y1 + y1^2
16x2^2 - 16x2x1 + 16y2^2 - 16y2y1 = x2^2 - 2x2x1 + x1^2 + y2^2 - 2y2y1 + y1^2
Canceling out terms:
15x2^2 + 15y2^2 = 15x1^2 + 15y1^2
Dividing by 15:
x2^2 + y2^2 = x1^2 + y1^2
This equation represents the line that contains the hypotenuse of RST.