Question
In the rectangular framework shown at right, GAC is a 40-degree angle, CAB is a 33-degree angle, segments AB and AD lie on the x-axis and y-axis, respectively, and AG = 10. Find the coordinates of G.
Answers
GPT 3.5
Since angle CAB is 33 degrees, angle DAG is 90 - 33 = 57 degrees.
This means triangle DAG is a 57-33-90 triangle, so AD = AG * cos(57) = 10 * cos(57).
Since angle GAC is 40 degrees, angle BAG is 180 - 33 - 40 = 107 degrees.
This means triangle BAG is a 33-107-40 triangle, so AB = AG * sin(107) = 10 * sin(107).
Since AB lies on the x-axis, the x-coordinate of G is AB = 10 * sin(107).
Since AD lies on the y-axis, the y-coordinate of G is -AD = -10 * cos(57).
Therefore, the coordinates of G are (10 * sin(107), -10 * cos(57)).
This means triangle DAG is a 57-33-90 triangle, so AD = AG * cos(57) = 10 * cos(57).
Since angle GAC is 40 degrees, angle BAG is 180 - 33 - 40 = 107 degrees.
This means triangle BAG is a 33-107-40 triangle, so AB = AG * sin(107) = 10 * sin(107).
Since AB lies on the x-axis, the x-coordinate of G is AB = 10 * sin(107).
Since AD lies on the y-axis, the y-coordinate of G is -AD = -10 * cos(57).
Therefore, the coordinates of G are (10 * sin(107), -10 * cos(57)).