In triangle GHI, if angle G = 30 degrees and side GH = 10 cm, what is the length of side HI if angle H = 60 degrees? (2023)

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Q: In triangle GHI, if angle G = 30 degrees and side GH = 10 cm, what is the length of side HI if angle H = 60 degrees? (2023)

Solution: Using the sine rule, HI/sin(60) = GH/sin(30). Therefore, HI = (10 * sin(60)) / sin(30) = 10 * (√3/2) / (1/2) = 10√3 cm.

Steps: 10

Step 1: Identify the given information in triangle GHI. We have angle G = 30 degrees, angle H = 60 degrees, and side GH = 10 cm.
Step 2: Calculate angle I using the fact that the sum of angles in a triangle is 180 degrees. Angle I = 180 - (angle G + angle H) = 180 - (30 + 60) = 90 degrees.
Step 3: Use the sine rule, which states that the ratio of a side to the sine of its opposite angle is the same for all sides in a triangle. The formula is: side_a/sin(angle_A) = side_b/sin(angle_B).
Step 4: Set up the sine rule for sides HI and GH. We have HI/sin(60) = GH/sin(30).
Step 5: Substitute the known values into the equation: HI/sin(60) = 10/sin(30).
Step 6: Calculate sin(60) and sin(30). We know sin(60) = √3/2 and sin(30) = 1/2.
Step 7: Substitute these values into the equation: HI/(√3/2) = 10/(1/2).
Step 8: Simplify the right side: 10/(1/2) = 10 * 2 = 20.
Step 9: Now we have HI/(√3/2) = 20. To find HI, multiply both sides by (√3/2): HI = 20 * (√3/2).
Step 10: Simplify the expression: HI = 20√3/2 = 10√3 cm.