Question
A coordinate plane with three lines graphed. A vertical line through
horizontal line through
and
horizontal line through
and
Ask by Boone Gross. in the United States
Jan 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The vertices of the triangular region are
Solution
To analyze the triangular region formed by the three lines on the coordinate plane, we need to identify the equations of the lines and the vertices of the triangle.
Step 1: Identify the lines
-
Vertical line through
: - This line is represented by the equation
.
- This line is represented by the equation
-
Horizontal line through
: - This line is represented by the equation
.
- This line is represented by the equation
-
Line through the points
and : - To find the equation of this line, we first calculate the slope
: - Using the point-slope form
with point :
- To find the equation of this line, we first calculate the slope
Step 2: Find the vertices of the triangle
The vertices of the triangle are the intersection points of the lines:
-
Intersection of
and : - Substituting
into gives the point .
- Substituting
-
Intersection of
and : - Substituting
into : - This gives the point
.
- Substituting
-
Intersection of
and : - Setting
: - This gives the point
.
- Setting
Step 3: List the vertices
The vertices of the triangle formed by the lines are:
Step 4: Identify the shaded region
The shaded triangular region is formed by the vertices
Conclusion
The vertices of the triangular region formed by the three lines are:
The points you provided
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Simplify this solution Extra Insights
Did you know that the concept of coordinate planes dates back to the 17th century? René Descartes introduced the Cartesian coordinate system, which revolutionized mathematics by allowing geometric shapes to be expressed in algebraic terms. This paved the way for analytic geometry and the ability to derive equations from shapes, enabling all sorts of advancements in science and engineering!
Now, let’s talk about that triangular region you’ve shaded! Triangles are not only fundamental shapes in geometry, but they also find real-world applications in architecture and construction. Stable structures, like bridges and roofs, often rely on triangular designs due to their strength and ability to distribute weight evenly. So, the next time you walk across a bridge, remember the power of triangles!
