What is the number of square units in the triangular region bounded by the lines y=2x, y=−2x, and x=5?

To find the area of the triangular region bounded by the lines \( y = 2x \), \( y = -2x \), and \( x = 5 \), we can follow these steps:

1. **Determine the vertices of the triangle:**

   - The line \( y = 2x \) intersects \( x = 5 \) at the point \( (5, 10) \).
   - The line \( y = -2x \) intersects \( x = 5 \) at the point \( (5, -10) \).
   - The two lines \( y = 2x \) and \( y = -2x \) intersect at the origin \( (0, 0) \).

   So, the vertices of the triangle are \( (0, 0) \), \( (5, 10) \), and \( (5, -10) \).

2. **Calculate the base and height of the triangle:**

   - The base of the triangle is the vertical distance between the points \( (5, 10) \) and \( (5, -10) \), which is \( 10 - (-10) = 20 \) units.
   - The height of the triangle is the horizontal distance from the origin \( (0, 0) \) to the line \( x = 5 \), which is 5 units.

3. **Use the formula for the area of a triangle:**

   The area \( A \) of a triangle is given by:

   \[
   A = \frac{1}{2} \times \text{base} \times \text{height}
   \]

   Substituting the values we found:

   \[
   A = \frac{1}{2} \times 20 \times 5 = 50 \text{ square units}
   \]

So, the number of square units in the triangular region is **50**.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do we find the area of a triangle given its vertices?
2. How does the slope of a line affect the shape of a triangle formed with it?
3. What is the significance of the intersection points of these lines?
4. How would the area change if the lines intersected at a different x-coordinate?
5. How can we generalize this problem to any set of linear equations forming a triangle?

**Tip:** The area of a triangle can be quickly found using the determinant method if the coordinates of its vertices are known.

Discover how to calculate the area of a triangle formed by the lines y=2x, y=-2x, and x=5 with a step-by-step solution. Learn about the triangle's vertices, base, height, and the formula used to find its area. Suitable for students in Grades 7-9.

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