Finding a Missing Vertex of a Polygon
To find a missing vertex of a polygon on the coordinate plane, use the properties of the shape. For a rectangle with known vertices A(2,1), B(7,1), and C(7,5), the fourth vertex D must have the same x-coordinate as A and the same y-coordinate as C, giving D(2,5). For a right triangle with legs parallel to the axes, the right-angle vertex shares its x with one given point and its y with the other: if P(-3,-2) and Q(4,6), then the right angle vertex is either (-3,6) or (4,-2). This geometric reasoning skill from enVision Mathematics, Grade 8, Chapter 7 connects coordinate geometry and shape properties.
Key Concepts
Use the properties of a geometric figure, such as parallel or perpendicular sides, to determine the coordinates of a missing vertex. For polygons with sides parallel to the axes, a missing vertex often shares an $x$ coordinate with one adjacent vertex and a $y$ coordinate with another.
Common Questions
How do I find a missing vertex of a rectangle on a coordinate plane?
A rectangle has opposite sides parallel. The missing vertex shares its x-coordinate with one adjacent known vertex and its y-coordinate with the other adjacent known vertex.
Rectangle has vertices A(2,1), B(7,1), C(7,5). Find vertex D.
D must align horizontally with A and vertically with C: D = (2, 5).
How do I find the right angle vertex of a right triangle with legs parallel to the axes?
The right angle vertex shares the x-coordinate with one given point and the y-coordinate with the other. There are two possible locations.
Points P(-3,-2) and Q(4,6) are vertices of a right triangle with axis-parallel legs. Find the right angle vertex.
The right angle vertex is either (-3,6) (shares x with P and y with Q) or (4,-2) (shares x with Q and y with P).
What property of rectangles makes it easy to find missing vertices?
In a rectangle, sides are parallel to each other. When sides align with the coordinate axes, opposite vertices share either x-coordinates or y-coordinates, making the missing vertex straightforward to calculate.
When do 8th graders learn to find missing polygon vertices?
Chapter 7 of enVision Mathematics, Grade 8 covers this in the Pythagorean Theorem unit.