Answer:
The length of side BC in triangle ABC is 5.525 cm, and the length of side ST in triangle RST is 0.3802x + 0.9505 cm.
Explanation:
Since angle A and angle R are both congruent to angle B, we know that angle A = angle R = y + 7°. We can use this information, along with the angles and sides of the triangles, to find the value of y and the lengths of the sides of the triangles.
From the given information, we know that the angles of triangle ABC add up to 180°, so we can use that to find the value of y:
4y - 71° + y + 7° + 106 - 2y° = 180°
6y - 64° = 180°
6y = 244°
y = 40.67°
We can use the value of y to find the measure of angle C:
C = 106 - 2y° = 106 - 2(40.67°) = 106 - 81.34° = 24.66°
Now that we know the measure of angle C, we can use the Law of Sines to find the length of side BC:
BC / sin C = AC / sin A
BC = (AC * sin C) / sin A
BC = (13 cm * sin 24.66°) / sin (4y - 71°)
BC = (13 cm * 0.4237) / sin (4(40.67°) - 71°)
BC = 5.525 cm
We can use the same method to find the length of side ST in triangle RST:
ST = (RT * sin C) / sin A
ST = ((2x + 5 cm) * sin 24.66°) / sin (4y - 71°)
ST = ((2x + 5 cm) * 0.4237) / sin (4(40.67°) - 71°)
ST = (2x + 5 cm) * 0.4237 * 0.4514
ST = (2x + 5 cm) * 0.1901
ST = 0.3802x + 0.9505 cm
Therefore, the length of side BC in triangle ABC is 5.525 cm, and the length of side ST in triangle RST is 0.3802x + 0.9505 cm.