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To calculate the height \( h \) of the triangle, we will use trigonometric principles and the provided information:

1. **Analyze the triangle**:
   - The triangle has an angle of \( 105^\circ \) and another angle \( 45^\circ \).
   - The side opposite the \( 105^\circ \) angle is \( 26 \, \text{cm} \).

2. **Apply the Sine Rule or other trigonometric identities**:
   Using the Sine Rule or basic geometry, we'll compute \( h \), the height perpendicular to the base.

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Let me compute this. math

# Given values
side_c = 26  # cm, side opposite to the 105° angle
angle_105 = 105  # degrees
angle_45 = 45  # degrees

# The third angle in the triangle
angle_30 = 180 - angle_105 - angle_45  # Triangle sum property

# Calculating height h using trigonometry
# h = side_c * sin(angle_45)
h = side_c * math.sin(math.radians(angle_45))

hThe height \( h \) of the triangle is approximately \( 18.385 \, \text{cm} \) when simplified to three decimal places.

Would you like a detailed explanation of the steps or further clarification? 

Here are five related questions you might find interesting:
1. How does the Sine Rule help in solving triangles like this one?
2. What are other ways to calculate height in non-right triangles?
3. How would you solve this problem if the given side and angles were different?
4. Can you explain the properties of angles in a triangle?
5. What is the significance of calculating the perpendicular height in real-world problems?

**Tip:** Always check if a triangle is right-angled before deciding on the approach, as it simplifies calculations.

Work out the height, h, of the triangle below. Give your answer in its simplest form.

Learn how to calculate the height of a triangle using trigonometric principles. This problem involves angles of 105°, 45°, and a side of 26 cm, applying the sine rule and geometry. Perfect for high school students in Grades 9-12.

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