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### Given Information:
- Triangle \( POQ \) is isosceles, with \( PO = OQ = 32 \, \text{cm} \).
- The perpendicular \( OM \) from \( O \) to the base \( PQ \) is 16 cm.

### (i) **Calculate the size of angle \( \angle P\hat{O}Q \):**

Since \( PO = OQ \), triangle \( POQ \) is isosceles, and \( OM \) is the perpendicular height from \( O \) to \( PQ \), it bisects \( PQ \). Therefore, triangle \( POM \) is a right triangle.

We can use trigonometry to find \( \angle P\hat{O}Q \). Specifically, we can find half of \( \angle P\hat{O}Q \), i.e., \( \angle POM \), using the tangent function:

\[
\tan(\angle POM) = \frac{PM}{OM}
\]

Since \( OM = 16 \, \text{cm} \) and \( PO = 32 \, \text{cm} \), we can find \( PM \) (half of \( PQ \)) using the Pythagorean theorem:

\[
PO^2 = PM^2 + OM^2
\]
\[
32^2 = PM^2 + 16^2
\]
\[
1024 = PM^2 + 256
\]
\[
PM^2 = 768
\]
\[
PM = \sqrt{768} \approx 27.71 \, \text{cm}
\]

Now we can calculate \( \tan(\angle POM) \):

\[
\tan(\angle POM) = \frac{PM}{OM} = \frac{27.71}{16} \approx 1.732
\]

Taking the arctangent to find \( \angle POM \):

\[
\angle POM = \tan^{-1}(1.732) \approx 60^\circ
\]

Thus, \( \angle P\hat{O}Q = 2 \times \angle POM = 2 \times 60^\circ = 120^\circ \).

### (ii) **Find the area of triangle \( POQ \):**

The area of triangle \( POQ \) can be calculated using the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, the base is \( PQ = 2 \times PM = 2 \times 27.71 = 55.42 \, \text{cm} \), and the height is \( OM = 16 \, \text{cm} \).

\[
\text{Area} = \frac{1}{2} \times 55.42 \times 16 \approx 443.36 \, \text{cm}^2
\]

### Final Answers:
1. \( \angle P\hat{O}Q = 120^\circ \)
2. The area of triangle \( POQ \) is approximately \( 443.36 \, \text{cm}^2 \).

Would you like further clarifications, or do you have any questions on this problem?

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**Relative Questions:**
1. How do you use the tangent function in right triangles?
2. What is the importance of the perpendicular bisector in an isosceles triangle?
3. How do you apply the Pythagorean theorem in right-angled triangles?
4. Can we use another method to find the area of an isosceles triangle without the height?
5. What are other ways to calculate the angle in a triangle besides trigonometry?

**Tip:** When working with isosceles triangles, symmetry is a useful property to simplify calculations, especially for finding angles and lengths!

Triangle POQ is an isosceles triangle. PO = OQ = 32 cm and OM = 16 cm. (i) Calculate the size of angle PÔQ. (ii) Find the area of triangle POQ.

Learn how to solve an isosceles triangle problem where PO = OQ = 32 cm and OM = 16 cm. Calculate angle PÔQ using trigonometric ratios and find the area of the triangle. Suitable for students in Grades 10-12.

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