https://file.aiquickdraw.com/mathbot/file/1725425874839ztd6ee5d.jpg

Based on the image, you have a right triangle with the following dimensions:

- The base \( PQ \) is 5.5 cm.
- The hypotenuse \( PQ \) is 5.3 cm.
- The height \( h \) is perpendicular to the base \( PQ \).

You are asked to determine:

i. The height \( h \).

ii. The angle \( \angle P \).

### Step 1: Determine the height \( h \)

We can use the Pythagorean theorem for this right triangle. The theorem states that:

\[
PQ^2 = h^2 + QP^2
\]

Where:

- \( PQ \) is the hypotenuse, which is 5.3 cm.
- \( QP \) is the base, which is 5.5 cm.
- \( h \) is the height we need to find.

However, from the diagram, it seems there's a typo in the lengths (since typically the hypotenuse is the longest side). Given that the hypotenuse (the longest side) should be longer than either of the other sides, we will adjust by assuming that \( PQ \) (5.5 cm) is the base, and 5.3 cm is the side opposite \( \angle P \).

### Step 2: Determine the angle \( \angle P \)

We can use trigonometric ratios. The sine of \( \angle P \) is given by:

\[
\sin \angle P = \frac{h}{PQ}
\]

And the cosine of \( \angle P \) is given by:

\[
\cos \angle P = \frac{adjacent \ side}{hypotenuse} = \frac{base}{hypotenuse} = \frac{PQ}{hypotenuse}
\]

Now, let me calculate the values for \( h \) and \( \angle P \).Based on the calculations:

i. The height \( h \) is approximately **1.47 cm**.

ii. The angle \( \angle P \) is approximately **15.5°**.

Would you like more details on the process, or do you have any other questions?

Here are some related questions to expand on this topic:

1. How can you verify if the triangle's sides form a right triangle?
2. What trigonometric function relates the height to the hypotenuse?
3. If the length of the hypotenuse were different, how would that affect the angle \( \angle P \)?
4. How would you calculate the other angle in the triangle?
5. How would the problem change if the triangle were not a right triangle?

**Tip:** Always double-check the dimensions provided in a problem, especially in geometric figures, to ensure they correspond to a valid configuration.

Based on Figure 1, determine:

i. The height (h)

ii. The angle ∠P.

This problem involves calculating the height and angle in a right triangle using the Pythagorean theorem and trigonometric ratios. The base is 5.5 cm, the hypotenuse is 5.3 cm, and the height is found to be approximately 1.47 cm. The angle ∠P is approximately 15.5°. Suitable for students in Grades 8-10.

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