1. **Problem 12:** For a 200" diagonal image with a display resolution of 1024 x 768 pixels, find the screen coordinates of points located on the LCD screen. 2. **Step 1:** Understand the problem. The image diagonal is 200 inches, and the resolution is 1024 pixels wide by 768 pixels high. 3. **Step 2:** Calculate the width and height of the image in inches using the aspect ratio 4:3 (since 1024:768 = 4:3). $$\text{Let width} = W, \quad \text{height} = H$$ $$\frac{W}{H} = \frac{4}{3}$$ $$W^2 + H^2 = 200^2 = 40000$$ 4. **Step 3:** Express $W$ in terms of $H$: $$W = \frac{4}{3}H$$ Substitute into the diagonal equation: $$\left(\frac{4}{3}H\right)^2 + H^2 = 40000$$ $$\frac{16}{9}H^2 + H^2 = 40000$$ $$\frac{25}{9}H^2 = 40000$$ 5. **Step 4:** Solve for $H$: $$H^2 = 40000 \times \frac{9}{25} = 14400$$ $$H = \sqrt{14400} = 120 \text{ inches}$$ 6. **Step 5:** Calculate $W$: $$W = \frac{4}{3} \times 120 = 160 \text{ inches}$$ 7. **Step 6:** Calculate the pixel size in inches: $$\text{Pixel width} = \frac{160}{1024} \approx 0.15625 \text{ inches/pixel}$$ $$\text{Pixel height} = \frac{120}{768} = 0.15625 \text{ inches/pixel}$$ 8. **Step 7:** To find the screen coordinates of a point given in inches, convert inches to pixels by dividing by pixel size and shifting origin from center to top-left (screen coordinates origin). $$x_{screen} = \frac{x_{inches} + \frac{W}{2}}{\text{pixel width}}$$ $$y_{screen} = \frac{\frac{H}{2} - y_{inches}}{\text{pixel height}}$$ 9. **Step 8:** Example: For a point at $(x,y)$ inches, calculate screen coordinates. --- 10. **Problem 13:** Given points on the screen: $$A' = (113, -40), B' = (72, 32), C' = (-12, 32), D' = (-53, -40), E' = (-12, -112), F' = (72, -112)$$ These are in inches relative to the center. 11. **Step 1:** The LEDs are placed on a circle of radius 100 inches at 60 degree intervals. 12. **Step 2:** Calculate coordinates of LEDs A-F using vector length 100 and angles $\theta = 0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$ respectively. 13. **Step 3:** Use formulas: $$x = r \cos \theta$$ $$y = r \sin \theta$$ where $r=100$ inches. 14. **Step 4:** Calculate each: - $A = (100 \cos 0^\circ, 100 \sin 0^\circ) = (100, 0)$ - $B = (100 \cos 60^\circ, 100 \sin 60^\circ) = (100 \times 0.5, 100 \times 0.866) = (50, 87)$ - $C = (100 \cos 120^\circ, 100 \sin 120^\circ) = (100 \times -0.5, 100 \times 0.866) = (-50, 87)$ - $D = (100 \cos 180^\circ, 100 \sin 180^\circ) = (-100, 0)$ - $E = (100 \cos 240^\circ, 100 \sin 240^\circ) = (100 \times -0.5, 100 \times -0.866) = (-50, -87)$ - $F = (100 \cos 300^\circ, 100 \sin 300^\circ) = (100 \times 0.5, 100 \times -0.866) = (50, -87)$ 15. **Step 5:** Round to nearest integer: $$A = (100, 0), B = (50, 87), C = (-50, 87), D = (-100, 0), E = (-50, -87), F = (50, -87)$$ --- **Final answers:** - For problem 12, use the conversion formulas in step 8 to find screen coordinates. - For problem 13, LED coordinates are: $$A = (100, 0), B = (50, 87), C = (-50, 87), D = (-100, 0), E = (-50, -87), F = (50, -87)$$