In triangle GHI, if angle G = 30 degrees and angle H = 60 degrees, what is the l

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Question: In triangle GHI, if angle G = 30 degrees and angle H = 60 degrees, what is the length of side GH if side GI = 10 cm?

Options:

Correct Answer: 8.66 cm

Solution:

Using the sine rule: GH/sin(30) = GI/sin(60). Therefore, GH = 10 * sin(30)/sin(60) = 10 * 0.5/(√3/2) = 10 * 1/√3 = 10/√3 ≈ 8.66 cm.

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

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

Step 1: Identify the angles in triangle GHI. We have angle G = 30 degrees and angle H = 60 degrees.
Step 2: Calculate angle I using the fact that the sum of angles in a triangle is 180 degrees. So, angle I = 180 - (30 + 60) = 90 degrees.
Step 3: Recognize that triangle GHI is a right triangle because angle I is 90 degrees.
Step 4: Use the sine rule, which states that the ratio of a side length to the sine of its opposite angle is constant. The formula is: GH/sin(G) = GI/sin(I).
Step 5: Substitute the known values into the sine rule: GH/sin(30) = 10/sin(90).
Step 6: Calculate sin(30) and sin(90). We know sin(30) = 0.5 and sin(90) = 1.
Step 7: Substitute these values into the equation: GH/0.5 = 10/1.
Step 8: Solve for GH by multiplying both sides by 0.5: GH = 10 * 0.5 = 5 cm.
Step 9: Since we need to find GH in terms of GI, we can also use the sine rule in another way: GH/sin(30) = GI/sin(60).
Step 10: Substitute GI = 10 cm and sin(60) = √3/2 into the equation: GH/sin(30) = 10/(√3/2).
Step 11: Rearrange to find GH: GH = 10 * sin(30)/(√3/2).
Step 12: Substitute sin(30) = 0.5 into the equation: GH = 10 * 0.5/(√3/2).
Step 13: Simplify the equation: GH = 10 * 0.5 * (2/√3) = 10/√3.
Step 14: Calculate the approximate value of GH: GH ≈ 8.66 cm.

Sine Rule – The sine rule relates the lengths of sides of a triangle to the sines of its angles, allowing for the calculation of unknown side lengths when certain angles and sides are known.
Triangle Angle Sum – The sum of the angles in a triangle is always 180 degrees, which can help in determining unknown angles.