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Proving Geometric Theorems about Circles and Parabolas

Learn how to use coordinates to prove or disprove statements involving circles and parabolas. Understand the concepts of theorems, equations, and properties of these geometric shapes.

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Proving Geometric Theorems about Circles and Parabolas

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  1. Introduction A theorem is statement that is shown to be true. Some important theorems have names, such as the Pythagorean Theorem, but many theorems do not have names. In this lesson, we will apply various geometric and algebraic concepts to prove and disprove statements involving circles and parabolas in a coordinate plane. If a statement is proven, it is a theorem. If a statement is disproved, it is not a theorem. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  2. Introduction, continued The directions for most problems will have the form “Prove or disprove…,” meaning we will work through those problems to discover whether each statement is true or false. Then, at the end of the work, we will state whether we have proved or disproved the statement. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  3. Key Concepts A theorem is any statement that is proven or can be proved to be true. The standard form of an equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2. This is based on the fact that any point (x, y) is on the circle if and only if 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  4. Key Concepts, continued Completing the square is the process of determining the value of and adding it to x2 + bx to form the perfect square trinomial A quadratic function can be represented by an equation of the form y = ax2 + bx + c, where a ≠ 0. If the function is named f, then its equation is 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  5. Key Concepts, continued The graph of any quadratic function is a parabola that opens up or down. A parabola is the set of all points that are equidistant from a fixed line, called the directrix, and a fixed point not on that line, called the focus. The parabola, directrix, and focus are all in the same plane. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  6. Key Concepts, continued The distance between the focus and a point on the parabola is the same as the distance from that point to the directrix. The vertex of the parabola is the point on the parabola that is closest to the directrix. Every parabola is symmetric about a line called the axis of symmetry. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  7. Key Concepts, continued The axis of symmetry intersects the parabola at the vertex. The x-coordinate of the vertex is The y-coordinate of the vertex is The standard form of an equation of a parabola that opens up or down and has vertex (h, k) is (x – h)2 = 4p(y – k), where p ≠ 0 and p is the distance between the vertex and the focus and between the vertex and the directrix. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  8. Key Concepts, continued Parabolas that open up or down represent functions, and their equations can be written in either of the following forms: y = ax2 + bx+ c or (x – h)2 = 4p(y – k). If one form is known, the other can be found. The standard form of an equation of a parabola that opens right or left and has vertex (h, k) is (y – k)2 = 4p(x – h), where p ≠ 0 and p is the distance between the vertex and the focus and between the vertex and the directrix. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  9. Key Concepts, continued In any parabola: The focus and directrix are each |p|units from the vertex. The focus and directrix are 2|p|units from each other. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  10. Common Errors/Misconceptions neglecting to square the radius for the equation of the circle using the equation of a parabola that opens up or down for a parabola that opens left or right, and vice versa 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  11. Guided Practice Example 1 Given the point A (–6, 0), prove or disprove that point A is on the circle centered at the origin and passing through 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  12. Guided Practice: Example 1, continued Draw a circle on a coordinate plane using the given information. You do not yet know if point A lies on the circle, so don’t include it in your diagram. In the diagram that follows, the name P is assigned to the origin and G is assigned to the known point on the circle. To help in plotting points, you can use a calculator to find decimal approximations. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  13. Guided Practice: Example 1, continued 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  14. Guided Practice: Example 1, continued Find the radius of the circle using the distance formula. Use the known points, P and G, to determine the radius of the circle. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  15. Guided Practice: Example 1, continued The radius of the circle is 6 units. For point A to be on the circle, it must be precisely 6 units away from the center of the circle. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  16. Guided Practice: Example 1, continued Find the distance of point A from the center P to determine whether it is on the circle. The coordinates of point P are (0, 0). The coordinates of point A are (–6, 0). 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  17. Guided Practice: Example 1, continued Point A is 6 units from the center, and since the radius of the circle is 6 units, point A is on the circle. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  18. Guided Practice: Example 1, continued The original statement has been proved, so it is a theorem. ✔ 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  19. Guided Practice: Example 1, continued 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  20. Guided Practice Example 2 Prove or disprove that the quadratic function graph with vertex (–4, 0) and passing through (0, 8) has its focus at (–4, 1). 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  21. Guided Practice: Example 2, continued Sketch the graph using the given information. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  22. Guided Practice: Example 2, continued Derive an equation of the parabola from its graph. The parabola opens up, so it represents a function. Therefore, its equation can be written in either of these forms: y = ax2 + bx + c or (x – h)2 = 4p(y – k). We were given a vertex and a point on the parabola; therefore, we’ll use the form (x – h)2 = 4p(y – k) and the vertex to begin deriving the equation. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  23. Guided Practice: Example 2, continued The vertex is (–4, 0), so h = –4 and k = 0. The equation of the parabola is (x + 4)2 = 4py. 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  24. Guided Practice: Example 2, continued Substitute the given point on the parabola into the standard form of the equation to solve for p. The point given is (0, 8). 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  25. Guided Practice: Example 2, continued Use the value of p to determine the focus. p is positive, so the focus is directly above the vertex. p = , so the focus is unit above the vertex. The vertex is (–4, 0), so the focus is 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  26. Guided Practice: Example 2, continued This result disproves the statement that the quadratic function graph with vertex (–4, 0) and passing through (0, 8) has its focus at (–4, 1). The statement has been disproved, so it is not a theorem. Instead, the following statement has been proved: The quadratic function graph with vertex (–4, 0) and passing through (0, 8) has its focus at 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  27. Guided Practice: Example 2, continued ✔ 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

  28. Guided Practice: Example 2, continued 6.7.1: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas

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