Math Problem Statement

Work out the height, h, of the triangle given a side of 34 cm, and angles of 105°, 45°, and 30°. Give your answer in its simplest form.

Solution

To find the height hh of the triangle, we can use trigonometric relationships within the given triangle.

Step-by-step solution:

  1. Given information:

    • The side of the triangle has a length of 34 cm.
    • The angle opposite the height hh is 4545^\circ.
    • The angle between the given side and the base is 105105^\circ.
  2. Using the sine rule (Law of Sines): In this triangle, the height hh forms a right triangle with the angle 4545^\circ at the bottom right. We can use the sine of the angle 4545^\circ to find the height hh.

    From the right triangle: sin(45)=h34\sin(45^\circ) = \frac{h}{34}

  3. Solving for hh: Using the known value for sin(45)=22\sin(45^\circ) = \frac{\sqrt{2}}{2}: 22=h34\frac{\sqrt{2}}{2} = \frac{h}{34}

    Multiply both sides by 34: h=34×22h = 34 \times \frac{\sqrt{2}}{2} h=172h = 17\sqrt{2}

Thus, the height hh is 172cm17\sqrt{2} \, \text{cm} in its simplest form.

Would you like more details or have any questions?

Related Questions:

  1. How is the sine rule different from the cosine rule?
  2. Can we find the other sides of this triangle using trigonometry?
  3. What is the significance of the angle 105105^\circ in this triangle?
  4. How would the height change if the angle at the bottom right was different?
  5. Could we solve this using the Law of Cosines instead?

Tip:

Always check if there is a simpler right-triangle approach for problems like this before using more complex laws like sine or cosine rules!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-angle triangle
Sine function

Formulas

sin(45°) = h / 34

Theorems

Sine rule
Pythagoras' theorem (implicitly)

Suitable Grade Level

Grades 9-11