In order to decide the most specific name for any quadrilateral, we need to look at the properties of the special quadrilaterals. The Quadrilateral Tree (hierarchy) will help.
Note that there are four quadrilaterals under the parallelogram branch of the tree. So if a quadrilateral has the properties of a parallelogram, is it only a parallelogram? Or is it something more specific, such as a rectangle, rhombus, or square?
Just as you learned with Prove Quads are Parallelograms, there are ways to prove the other quadrilaterals using their properties:
Just as you learned with Prove Quads are Parallelograms, there are ways to prove the other quadrilaterals using their properties:
Rectangle |
A quadrilateral that is a parallelogram with
|
Rhombus |
A quadrilateral that is a parallelogram with
|
Square |
A quadrilateral that is a parallelogram with:
|
Kite |
A quadrilateral with:
|
Trapezoid |
A quadrilateral with:
|
Isosceles Trapezoid |
A quadrilateral that is a trapezoid with:
|
Example 2:
What is the most specific name for this quadrilateral? Solution: Looking at this quadrilateral it is also tempting to say the figure is a square because it looks like it. Analyzing what is shown on the figure is two angles that are congruent to each other. These angles are congruent alternate interior angles which means that two of the segments are parallel. So we can add parallel markings to the diagram. Therefore the most specific name for this quadrilateral is a trapezoid. The figure is not an isosceles trapezoid because it does not have legs that are shown as congruent. |
Coordinate Plane:
What can we determine about a quadrilateral on the coordinate plane?
Let’s take a new look at the Quadrilateral Tree and how we can use the information a coordinate plane gives us to decide the most specific name for any given quadrilateral:
What can we determine about a quadrilateral on the coordinate plane?
- Slopes of lines
- Same slope - parallel segments
- Opposite reciprocal slopes - perpendicular line segments and therefore there is a right angle
- Example: slopes of \(\large\frac{2}{3}\) and \(\large-\frac{3}{2}\) are opposite reciprocals.
- Neither - the line segments are neither parallel nor perpendicular
- Lengths of line segments (distances between points)
- Whether or not line segments are congruent
- Use the distance formula or Pythagorean Theorem to find lengths of line segments
- Midpoints
- Whether or not one or both diagonals are bisected (same midpoint)
Let’s take a new look at the Quadrilateral Tree and how we can use the information a coordinate plane gives us to decide the most specific name for any given quadrilateral:
First decide which branch of the quadrilateral tree your quadrilateral falls: parallelogram branch, kite, or trapezoid branch. Next, look at more properties to see which specific quadrilateral it is.
Another option is to draw the diagonals of your quadrilateral and check their properties to help you specify your quadrilateral. You can check if diagonals are perpendicular, congruent, bisect each other, or if only one diagonal is bisected.
Watch the videos which walk through finding the most specific name of the following quadrilaterals (named after great mathematicians). Then, you can try some on your own using the Quick Check. You may graph by hand, go to GeoGebra or Desmos and enter in the coordinates of each point.
Another option is to draw the diagonals of your quadrilateral and check their properties to help you specify your quadrilateral. You can check if diagonals are perpendicular, congruent, bisect each other, or if only one diagonal is bisected.
Watch the videos which walk through finding the most specific name of the following quadrilaterals (named after great mathematicians). Then, you can try some on your own using the Quick Check. You may graph by hand, go to GeoGebra or Desmos and enter in the coordinates of each point.
Example 3:
What is the most specific name for this quadrilateral? Solution: The figure looks like it might be an isosceles trapezoid. Let's use coordinates to see if we can justify our conjecture. Using the slope formula the slope of \(\overline{HA}=\large\frac{-2}{1}\) and the slope of \(\overline{WT}=\large\frac{-10}{5}\). Since the slopes are equal \(\overline{WT}\parallel\overline{HA}\). Using the distance formula \(WH=\sqrt{(-2-5)^2+(3-1)^2}=\sqrt{40}\) and \(AT= \sqrt{(5-3)^2+(-1--7)^2}=\sqrt{40}\). This means that \(\overline{WH}\cong\overline{AT}\). This verifies that \(WHAT\) is an isosceles trapezoid. |
Example 4:
What is the most specific name for each quadrilateral?
What is the most specific name for each quadrilateral?
Euclid
\(A\; (1, 0)\) \(B\; (3, 4)\) \(C\; (7, 6)\) \(D\; (5, 2)\) |
Aryabhata
\(J\: (-4, 2)\) \(K\; (1, -3)\) \(L\; (0, 4)\) \(M\; (-4, 7)\) |
Banneker
\(O\; (8, 1)\) \(P\; (13, 0)\) \(Q\; (4, -6)\) \(R\; (5, -1)\) |
Solution:
Watch the videos for the solutions
Watch the videos for the solutions
|
|
Quick Check
1) What is the most specific name for this quadrilateral?
Newton
\(A\; (1, 0)\)
\(B\; (2, 4)\)
\(C\; (7, 7)\)
\(D\; (6, 3)\)
2) What is the most specific name for this quadrilateral?
Archimedes
\(A\; (-9, 2)\)
\(B\; (-8, -1)\)
\(C\; (-3, 4)\)
\(D\; (-2, 1)\)
3) What is the most specific name for this quadrilateral?
Gauss
\(A\; (0, 2)\)
\(B\; (4, 4)\)
\(C\; (2, 8)\)
\(D\; (-2, 6)\)
4) What is the most specific name for this quadrilateral?
Pythagoras
\(A\; (1,5)\)
\(B\; (4, 5)\)
\(C\; (5, 2)\)
\(D\; (1, -1)\)
5) What is the most specific name for this quadrilateral?
Germain
\(A\; (2, 1)\)
\(B\; (6, 0)\)
\(C\; (2, 5)\)
\(D\; (10, 3)\_
Quick Check Solutions
1) What is the most specific name for this quadrilateral?
Newton
\(A\; (1, 0)\)
\(B\; (2, 4)\)
\(C\; (7, 7)\)
\(D\; (6, 3)\)
2) What is the most specific name for this quadrilateral?
Archimedes
\(A\; (-9, 2)\)
\(B\; (-8, -1)\)
\(C\; (-3, 4)\)
\(D\; (-2, 1)\)
3) What is the most specific name for this quadrilateral?
Gauss
\(A\; (0, 2)\)
\(B\; (4, 4)\)
\(C\; (2, 8)\)
\(D\; (-2, 6)\)
4) What is the most specific name for this quadrilateral?
Pythagoras
\(A\; (1,5)\)
\(B\; (4, 5)\)
\(C\; (5, 2)\)
\(D\; (1, -1)\)
5) What is the most specific name for this quadrilateral?
Germain
\(A\; (2, 1)\)
\(B\; (6, 0)\)
\(C\; (2, 5)\)
\(D\; (10, 3)\_
Quick Check Solutions