Given that PRS = 90^, and from the figure, PRQ = 90^.

You're on a roll — here are the solutions: 5. In the figure below angles QPR and PSR are equal. Angle PRS is a right angle. The ratio of PR:QR = 4:3. Given that the area of triangle PQR is 24 cm^2, find the area of triangle PRS. Step 1: Identify similar triangles. Given that PRS = 90^, and from the figure, PRQ = 90^. Given QPR = PSR. Since the sum of angles in a triangle is 180^: In PQR, PQR = 180^ - 90^ - QPR = 90^ - QPR. In PRS, RPS = 180^ - 90^ - PSR = 90^ - PSR. Since QPR = PSR, it follows that PQR = RPS. Therefore, PQR is similar to SPR (Angle-Angle-Angle similarity). Step 2: Use the given area of PQR and the ratio PR:QR to find the lengths of PR and QR. Let PR = 4k and QR = 3k for some constant k. The area of PQR = (1)/(2) × base × height = (1)/(2) × QR × PR. 24 = (1)/(2) × (3k) × (4k) 24 = (1)/(2) × 12k^2 24 = 6k^2 k^2 = 4 k = 2 So, PR = 4 × 2 = 8 cm and QR = 3 × 2 = 6 cm. Step 3: Use the property of similar triangles that the ratio of their areas is equal to the square of the ratio of their corresponding sides. Since PQR SPR, the ratio of their areas is: Area( PQR)Area( SPR) = ((QR)/(PR))^2 Using the sides QR=6 and PR=8: (24)/(Area)( SPR) = ((6)/(8))^2 (24)/(Area)( SPR) = ((3)/(4))^2 (24)/(Area)( SPR) = (9)/(16) Step 4: Solve for the area of PRS. Area( SPR) = 24 × (16)/(9) Area( SPR) = (8 × 3 × 16)/(3 × 3) Area( SPR) = (8 × 16)/(3) Area( SPR) = (128)/(3) Area( SPR) = 42 (2)/(3) cm^2 The area of triangle PRS is 42 (2)/(3) cm^2. 6. A suit whose marked price is sh. 8,000 is sold to customer after allowing him a discount of 13%. If the trader makes a profit of 20%, find how much the trader paid for the suit. Step 1: Calculate the selling price (SP) after the discount. Marked Price (MP) = Sh. 8,000. Discount = 13%. Discount amount = 13\% of 8,000 = 0.13 × 8,000 = Sh. 1,040. Selling Price (SP) = MP - Discount amount = 8,000 - 1,040 = Sh. 6,960. Alternatively, SP = MP × (1 - discount percentage) = 8,000 × (1 - 0.13) = 8,000 × 0.87 = Sh. 6,960. Step 2: Calculate the cost price (CP) using the profit percentage. Profit = 20%. The selling price (SP) is the cost price plus the profit. SP = CP + (20\% of CP) = CP × (1 + 0.20) = 1.20 × CP. 6,960 = 1.20 × CP CP = (6,960)/(1.20) CP = (6,960)/(120)100 = (6,960 × 100)/(120) CP = (696,000)/(120) = (69,600)/(12) CP = 5,800 The trader paid Sh. 5,800 for the suit. **7. Evans is a salesperson with Ajab Millers. He is paid on monthly basis as the agreement; basic pay of Ksh. 20,000, a commission