1. Problem 45: Given secants FA and EA intersecting outside the circle at A, with FA = 24 cm, LA = 10 cm, and MA = 12 cm, find EA. 2. Use the secant-secant power theorem: If two secants intersect outside a circle, then the product of the whole secant segment and its external segment is equal for both secants. 3. The formula is: $$FA \times LA = EA \times MA$$ 4. Substitute the known values: $$24 \times 10 = EA \times 12$$ 5. Simplify: $$240 = 12 \times EA$$ 6. Solve for EA: $$EA = \frac{240}{12} = 20$$ 7. Therefore, EA = 20 cm. --- 8. Problem 46: In circle Y with radius 12 cm and arc ES measuring 20°, find the area of the shaded sector in terms of $\pi$. 9. The area of a sector is given by: $$Area = \frac{\theta}{360} \times \pi r^{2}$$ where $\theta$ is the central angle in degrees. 10. Substitute values: $$Area = \frac{20}{360} \times \pi \times 12^{2} = \frac{20}{360} \times \pi \times 144$$ 11. Simplify: $$Area = \frac{1}{18} \times 144 \pi = 8 \pi$$ 12. The area of the shaded region is $8\pi$ cm². --- 13. Problem 47: A circular board with diameter 40 cm is cut into 20 congruent sectors. Find the area of each sector. 14. Radius $r = \frac{40}{2} = 20$ cm. 15. Total area of the circle: $$\pi r^{2} = \pi \times 20^{2} = 400 \pi$$ 16. Each sector area: $$\frac{400 \pi}{20} = 20 \pi$$ 17. Each sector has area $20\pi$ cm². --- 18. Problem 48: Coordinates of point L in isosceles trapezoid LIVE are given as options. 19. Given points I at (c, e), V at (d, 0), and L at (d-c, e) (from the graph description). 20. Therefore, the coordinates of L are $(d-c, e)$. --- 21. Problem 49: Determine the type of triangle with vertices (0,4), (-4,-2), and (4,-2). 22. Calculate side lengths using distance formula: - Between (0,4) and (-4,-2): $$\sqrt{(0+4)^2 + (4+2)^2} = \sqrt{16 + 36} = \sqrt{52}$$ - Between (0,4) and (4,-2): $$\sqrt{(0-4)^2 + (4+2)^2} = \sqrt{16 + 36} = \sqrt{52}$$ - Between (-4,-2) and (4,-2): $$\sqrt{( -4 - 4)^2 + (-2 + 2)^2} = \sqrt{64 + 0} = 8$$ 23. Two sides are equal ($\sqrt{52}$), so the triangle is isosceles. --- 24. Problem 50: Which equation represents a circle? 25. The general equation of a circle is $$x^{2} + y^{2} + Dx + Ey + F = 0$$ with both $x^{2}$ and $y^{2}$ terms having coefficient 1. 26. Option D: $$x^{2} + y^{2} = 25$$ matches the circle equation with center at (0,0) and radius 5. 27. Final answers: - 45: 20 cm - 46: 8\pi cm² - 47: 20\pi cm² - 48: (d-c, e) - 49: Isosceles triangle - 50: D