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The given triangle is isosceles, with two equal sides measuring 9 cm and a vertex angle of \(50^\circ\). We are tasked with finding the missing vertical height \(d\), which represents the perpendicular distance from the base to the top vertex.

### Steps to solve for \(d\):
1. **Divide the triangle into two right-angled triangles**:
   The isosceles triangle can be split into two identical right-angled triangles, each with a base half the length of the full base of the original triangle. 

2. **Identify known values**:
   - The length of the hypotenuse (the side of the original isosceles triangle) is 9 cm.
   - The angle at the top of each right-angled triangle is \(25^\circ\) (since \(50^\circ\) is divided in half).
   - \(d\) is the opposite side to the \(25^\circ\) angle.

3. **Apply trigonometry**:
   Use the sine function to find the height \(d\):
   \[
   \sin(25^\circ) = \frac{d}{9}
   \]
   Solving for \(d\):
   \[
   d = 9 \times \sin(25^\circ)
   \]

Let me calculate this.The missing height \(d\) is approximately \(3.80 \, \text{cm}\).

Would you like any further details or clarifications? Here are some related questions for further practice:

1. What formula is used for calculating the length of a side using sine in a right-angled triangle?
2. How would you calculate the base of the isosceles triangle using trigonometry?
3. What would happen if the vertex angle were larger, for instance, \(60^\circ\)?
4. How do you find the area of the isosceles triangle once you know the height?
5. How would you use cosine instead of sine to find the height \(d\)?

**Tip**: Always remember to split isosceles triangles into two right triangles when solving for unknowns using trigonometry!

In each triangle, find the missing length indicated by a letter. (The given triangle is isosceles with a vertex angle of 50°, two equal sides of 9 cm, and the missing height d.)

Learn how to calculate the missing height (d) in an isosceles triangle with a 50° vertex angle and two 9 cm equal sides using trigonometry. Split the triangle into two right-angled triangles and apply the sine function for a step-by-step solution. Suitable for Grades 9-10.

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