in the figure, PQR=PSR=90 degree, angle P in QPR=27.6 degree, angle T in PTS=63.9 degree. PR=5.7cm, PS=3.2cm, and PT=2.7cm. Calculate: a. QR b. SPR c. PST

Let's break down the problem step by step. We are given the following information about a geometric configuration:

1. \( \angle PQR = \angle PSR = 90^\circ \)
2. \( \angle P \) in \( \triangle QPR = 27.6^\circ \)
3. \( \angle T \) in \( \triangle PTS = 63.9^\circ \)
4. \( PR = 5.7 \, \text{cm} \)
5. \( PS = 3.2 \, \text{cm} \)
6. \( PT = 2.7 \, \text{cm} \)

We need to calculate:
- a. The length of \( QR \)
- b. The measure of \( \angle SPR \)
- c. The measure of \( \angle PST \)

### Step 1: Solving for \( QR \)
Using the right triangle \( \triangle PQR \), we can apply the trigonometric relationships.

- We know that \( \angle QPR = 27.6^\circ \) and \( PR = 5.7 \, \text{cm} \).
  
Using the sine function:
\[
\sin(\angle QPR) = \frac{QR}{PR}
\]
\[
QR = PR \times \sin(\angle QPR)
\]
Substituting the known values:
\[
QR = 5.7 \times \sin(27.6^\circ)
\]

### Step 2: Solving for \( \angle SPR \)
In the right triangle \( \triangle PSR \), we can calculate \( \angle SPR \) using trigonometric relationships.

Since \( \angle PSR = 90^\circ \), we can use:
\[
\sin(\angle SPR) = \frac{PS}{PR}
\]
Thus,
\[
\angle SPR = \arcsin\left(\frac{PS}{PR}\right)
\]
Substituting the known values:
\[
\angle SPR = \arcsin\left(\frac{3.2}{5.7}\right)
\]

### Step 3: Solving for \( \angle PST \)
In \( \triangle PST \), we are given that \( \angle T = 63.9^\circ \).

To find \( \angle PST \), use the fact that the sum of angles in a triangle is \( 180^\circ \):
\[
\angle PST = 180^\circ - \angle T - \angle PTS
\]
But first, we need to find \( \angle PTS \), which can be calculated using trigonometry in the right triangle \( \triangle PTS \).

Now, I'll calculate these values.Here are the calculated values:

- **a. Length of \( QR \)**: \( 2.64 \, \text{cm} \)
- **b. Measure of \( \angle SPR \)**: \( 34.15^\circ \)
- **c. Measure of \( \angle PST \)**: \( 90^\circ \)

This indicates that \( \triangle PST \) is a right triangle since \( \angle PST = 90^\circ \).

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

Here are 5 related questions:
1. How would you calculate the area of triangle \( \triangle PQR \)?
2. Can you find the length of \( SR \) in triangle \( \triangle PSR \)?
3. How would the calculations change if angle \( P \) was different in \( \triangle QPR \)?
4. What is the perimeter of triangle \( \triangle PTS \)?
5. How can the Pythagorean theorem be applied to solve for missing sides in these triangles?

**Tip:** In problems involving right triangles, trigonometric ratios (sine, cosine, tangent) are very helpful for calculating unknown sides and angles.

Explore how to solve a geometry problem involving right triangles and trigonometric functions. Calculate the lengths of sides and measures of angles using the sine function and the sum of angles theorem. Detailed step-by-step solution provided.

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