Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe the intersections as a segment and name the coordinates of the endpoints.
Lines That Pass Through Regions
How can we use the Pythagorean theorem to find the length of 𝐴𝐵, or in other words, the distance between 𝐴(−2,1) and 𝐵(3,3)? Find the distance between 𝐴 and 𝐵.
Consider the rectangular region:
a. Does a line of slope 2 passing through the origin intersect this rectangular region? If so, which boundary points of the rectangle does it intersect? Explain how you know.
b. Does a line of slope 1/2 passing through the origin intersect this rectangular region? If so, which boundary points of the rectangle does it intersect?
c. Does a line of slope 1/3 passing through the origin intersect this rectangular region? If so, which boundary points of the rectangle does it intersect?
d. A line passes through the origin and the lower right vertex of the rectangle. Does the line pass through the interior of the rectangular region or the boundary of the rectangular region? Does the line pass through both?
e. For which values of 𝑚 would a line of slope 𝑚 through the origin intersect this region?
f. For which values of 𝑚 would a line of slope 𝑚 through the point (0,1) intersect this region?
Consider the triangular region in the plane given by the triangle with vertices
𝐴(0,0), 𝐵(2,6), and 𝐶(4,2).
a. The horizontal line 𝑦 = 2 intersects this region. What are the coordinates of the two boundary points it intersects? What is the length of the horizontal segment within the region along this line?
b. Graph the line 3𝑥 −2𝑦 = 5. Find the points of intersection with the boundary of the triangular region, and label them as 𝑋 and 𝑌.
c. What is the length of the 𝑋𝑌?
d. A robot starts at position (1,3) and moves vertically downward toward the 𝑥-axis at a constant speed of 0.2 units per second. When will it hit the lower boundary of the triangular region that falls in its vertical path?
Consider the given rectangular region:
a. Draw lines that pass through the origin and through each of the vertices of the rectangular region. Do each of
the four lines cross multiple points in the region? Explain.
b. Write the equation of a line that does not intersect the rectangular region at all.
c. A robot is positioned at 𝐷 and begins to move in a straight line with slope 𝑚 = 1. When it intersects with a boundary, it then reorients itself and begins to move in a straight line with a slope of 𝑚 = −1/2. What is the location of the next intersection the robot makes with the boundary of the rectangular region?
d. What is the approximate distance of the robot’s path in part (c)?
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