Math Problem Statement

Using the previous information (To find the coordinates of the points where the two curves intersect, we need to set the two equations equal to each other since they intersect at those points. Given: (y = 2x² - 8x + 5 ) ---(1) (y = x² - 4x + 2) ---(2) Setting equation (1) equal to equation (2) gives: (2x² - 8x + 5 = x² - 4x + 2) Now, we need to solve this quadratic equation to find the x-values of the points of intersection. Subtracting (x² - 4x + 2) from both sides, we get: (2x² - x² - 8x + 4x + 5 - 2 = 0) (x² - 4x + 3 = 0) To solve this quadratic equation, we can either factorize it or use the quadratic formula. Factoring is easier in this case: ((x - 3)(x - 1) = 0) This gives us x = 3 or x = 1 as the x-coordinates of the points of intersection. To find the y-coordinates, we substitute these x-values back into either of the original equations. Let's use equation (2): For x = 3: (y = (3)² - 4(3) + 2) (y = 9 - 12 + 2) (y = -1) So, one point of intersection is (3, -1). For x = 1: (y = (1)² - 4(1) + 2) (y = 1 - 4 + 2) (y = -1) The other point of intersection is (1, -1) ) answer the following : If O is the origin, find the area of of the triangle OAB.

Solution

To find the area of triangle OAB, where O is the origin (0, 0) and A(3, -1) and B(1, -1) are the points of intersection, we can use the formula for the area of a triangle given by the coordinates of its vertices:

Area=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|

For triangle OAB, we have:

  • O(0,0)O(0, 0) → (x1,y1)=(0,0)(x_1, y_1) = (0, 0)
  • A(3,−1)A(3, -1) → (x2,y2)=(3,−1)(x_2, y_2) = (3, -1)
  • B(1,−1)B(1, -1) → (x3,y3)=(1,−1)(x_3, y_3) = (1, -1)

Substituting these coordinates into the area formula:

Area=12∣0(−1−(−1))+3(−1−0)+1(0−(−1))∣\text{Area} = \frac{1}{2} \left| 0(-1 - (-1)) + 3(-1 - 0) + 1(0 - (-1)) \right|

Calculating step by step:

  1. First term: 0×(−1+1)=00 \times (-1 + 1) = 0
  2. Second term: 3×(−1)=−33 \times (-1) = -3
  3. Third term: 1×1=11 \times 1 = 1

Now, substitute these results back into the formula:

Area=12∣0−3+1∣=12∣−2∣=12×2=1\text{Area} = \frac{1}{2} \left| 0 - 3 + 1 \right| = \frac{1}{2} \left| -2 \right| = \frac{1}{2} \times 2 = 1

Thus, the area of triangle OAB is 1\boxed{1}.

Additional Questions

  1. What are the implications of the area being 1 in terms of triangle shape?
  2. How would the area change if point A or B were shifted?
  3. What is the formula for the area of a triangle in general?
  4. Can you derive the area using the base-height method instead?
  5. How would you find the coordinates of the centroid of triangle OAB?

Tip

Always double-check the coordinates when calculating areas, as small mistakes can lead to incorrect results.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Triangle Area

Formulas

Area of a triangle given vertices: A = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Theorems

Triangle Area Theorem

Suitable Grade Level

Grades 9-12

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