Using only a simple edge and a square, make the following geometric designs, making the design lines clear.

(i) Construct the triangle PQR in which PQ = 7.0 cm, POR = 45 degress, QPR = 60degress.     (ii) Construct the angle bisector RPQ.

(iii) Name the point S where the parallel line drawn through R parallel to PQ meets the angle bisector.

(iv) Construct the triangle whose square product is equal to the square PQSR and one leg PR and name it PRT.

To complete this construction, you'll follow a series of geometric steps using a ruler (for measuring and drawing straight lines) and a protractor (for measuring angles). Here’s a step-by-step guide to creating the geometric design and solving the problem:

### **Step (i): Construct the Triangle PQR**

1. **Draw Line Segment PQ:**
   - Use your ruler to draw a straight line PQ with a length of 7.0 cm.

2. **Construct Angle POR = 45°:**
   - Place the protractor at point P and measure an angle of 45° above PQ. Mark the direction of this angle with a point O, but do not draw the full line yet.

3. **Construct Angle QPR = 60°:**
   - Place the protractor at point Q and measure an angle of 60° below PQ. Extend this line until it intersects with the line drawn from point P (the line passing through O).
   
4. **Mark the Intersection as R:**
   - The point where these two lines intersect is point R. Now you have your triangle PQR.

### **Step (ii): Construct the Angle Bisector of ∠RPQ**

1. **Bisect Angle ∠RPQ:**
   - Use a compass to construct the angle bisector. Place the compass at point P, draw an arc that crosses both PR and PQ. 
   - Without changing the compass width, place it on the intersections of the arc with PR and PQ, and draw two small arcs that intersect each other. 
   - Draw a straight line from P through the intersection of the small arcs. This line is the angle bisector of ∠RPQ.

### **Step (iii): Construct a Line Parallel to PQ through R and Name the Intersection Point S**

1. **Construct a Parallel Line to PQ:**
   - Place your set square on line PQ and align it carefully.
   - Slide the set square to point R while maintaining the alignment. Draw a straight line through R that is parallel to PQ.

2. **Mark the Intersection as Point S:**
   - The point where this parallel line intersects the angle bisector of ∠RPQ is point S.

### **Step (iv): Construct Triangle PRT Such That the Square of Its Area is Equal to the Area of Quadrilateral PQSR**

1. **Construct the Triangle PRT:**
   - First, calculate or approximate the area of quadrilateral PQSR (using a known method like the Shoelace theorem, or by dividing it into triangles).
   - From this area, deduce the length of the required side of triangle PRT.
   - Draw the leg PR as a side of this triangle.
   - Then, using an appropriate method (such as the cosine rule or by determining lengths using a given square root value), construct the remaining side PT such that the triangle PRT satisfies the given area condition.
  
   **Note:** The detailed construction of triangle PRT depends on the specific geometry and algebraic steps needed to ensure the area condition.

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### **Final Diagram Summary**
- Triangle PQR is constructed with given side lengths and angles.
- Angle bisector of ∠RPQ is drawn.
- Parallel line through R to PQ is drawn and intersects the angle bisector at S.
- Triangle PRT is constructed, such that its square area equals that of quadrilateral PQSR.

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Do you want more details or have any questions about the steps? Here are five related questions:

1. How do you calculate the area of quadrilateral PQSR?
2. What is the significance of the angle bisector in geometric constructions?
3. How can the length of the sides of triangle PRT be determined precisely?
4. What geometric principles ensure the parallel line construction is accurate?
5. How can you use the properties of similar triangles in this construction?

**Tip:** When constructing angles, double-check the measurements with a protractor to ensure accuracy.

Using only a simple edge and a square, make the following geometric designs, making the design lines clear. (i) Construct the triangle PQR in which PQ = 7.0 cm, POR = 45 degrees, QPR = 60 degrees. (ii) Construct the angle bisector RPQ. (iii) Name the point S where the parallel line drawn through R parallel to PQ meets the angle bisector. (iv) Construct the triangle whose square product is equal to the square PQSR and one leg PR and name it PRT.

Learn how to construct the triangle PQR with specified side lengths and angles, and its angle bisector RPQ. Discover the point S where a parallel line through R intersects the angle bisector. Construct triangle PRT such that its square area equals that of quadrilateral PQSR. Suitable for high school students studying advanced geometry.

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