Math Problem Statement

find coordinate of point E

Find the coordinate of point E where the diagonals of the rectangle intersect.

Solution

To find the coordinates of point EE, which is the intersection point of the diagonals of the rectangle, we can use the property that the diagonals of a rectangle bisect each other. Therefore, point EE will be the midpoint of the diagonals, which means it is the midpoint of both AC‾\overline{AC} and BD‾\overline{BD}.

Given coordinates:

  • A(−3,4)A(-3, 4)
  • B(9,10)B(9, 10)
  • D(1,−4)D(1, -4)
  • C(x,2)C(x, 2) — we do not know the value of xx, but can find it using midpoint properties.

The midpoint EE of diagonal AC‾\overline{AC} and diagonal BD‾\overline{BD} should be the same. Let's first calculate the midpoint of BD‾\overline{BD}.

Midpoint formula:

The midpoint of a segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Step 1: Midpoint of BD‾\overline{BD}

Using the coordinates B(9,10)B(9, 10) and D(1,−4)D(1, -4), we calculate the midpoint EE as follows:

= \left( \frac{10}{2}, \frac{6}{2} \right) = (5, 3)$$ #### Step 2: Midpoint of $$\overline{AC}$$ Now, we know that point $$E(5, 3)$$ is also the midpoint of $$\overline{AC}$$. Using the coordinates $$A(-3, 4)$$ and $$C(x, 2)$$, we can set up the midpoint equation: $$E = \left( \frac{-3 + x}{2}, \frac{4 + 2}{2} \right) = (5, 3)$$ From the second coordinate equation: $$\frac{4 + 2}{2} = 3 \quad \text{(This is true, so no need to solve further for the y-coordinate)}$$ Now, solving for the x-coordinate: $$\frac{-3 + x}{2} = 5$$ Multiplying both sides by 2: $$-3 + x = 10$$ Solving for $$x$$: $$x = 13$$ Thus, the coordinates of point $$C$$ are $$C(13, 2)$$. ### Final Answer: The coordinates of point $$E$$, where the diagonals intersect, are $$\boxed{(5, 3)}$$. Would you like further details on this solution, or have any other questions? Here are some related questions to explore: 1. How can we verify that the diagonals bisect each other in a rectangle? 2. What other properties of rectangles are important in coordinate geometry? 3. How can you find the equation of a diagonal using two points? 4. How do you calculate the area of a rectangle given the vertices? 5. What is the significance of midpoints in different types of quadrilaterals? **Tip:** Remember that in any parallelogram, including rectangles, the diagonals always bisect each other.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Rectangles
Midpoint Formula

Formulas

Midpoint formula: Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Theorems

Diagonals of a rectangle bisect each other

Suitable Grade Level

Grades 8-10

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