Embedded Files
Materials
Materials
Height worksheet, pages 2/3 (one per person)
Peg board and rubber bands (optional, one set per pair)
Straight edge (optional, one per pair)
Shearing
Shearing
Shearing is a transformation in the plan that moves points in an object in such a way that the area does not change. For a triangle, shearing can be accomplished by translating the vertex opposite the base (called the apex) along the line parallel to the base to form a new triangle. This new triangle has the same area as the original because
the two triangles share a base,
the new apex is on a line parallel to the base through the original apex, and
the perpendicular distance between parallel lines is constant, meaning the heights of both triangles are equal.
Students should be easily convinced that the perpendicular distance between parallel lines is constant, especially when the perpendicular distance is vertical or horizontal, as in the first four problems on the worksheet.
On an 11x11 pegboard, there are 10 possible shears of the apex that will result in a new triangle with the same area as the original when the base is aligned vertically or horizontally with the sides of the board.
We can find the area of a triangle by multiplying the length of the base and the height and dividing by two (see the Area Formulas activity). If students are struggling to understand why they need to divide by two, they can use their rubber bands to form quadrilaterals to see why the area of the triangle is half the area of a parallelogram.
When the height of an obtuse triangle does not pass through the interior of a triangle, it can be harder for students to calculate the area due to the difficulty in identifying the height. However, sometimes it can be easier for a student to see the area of an obtuse triangle as a difference between two right triangles with legs formed by the base and extension and the other leg by the height.
In the second two problems, we can find the area of the shaded triangle by taking the difference between a larger triangle (outlined in yellow) that includes the shaded portion and then subtracting the area of the unshaded right triangle formed by the base extension and the height. Seeing that these methods are equivalent can help a student be convinced that the area formula still works for obtuse triangles as long as the height has been identified correctly.
Our last two triangles are more challenging because the base and height do not align vertically or horizontally on the board. This means finding a line through the apex parallel to the base is going to be more challenging. The first challenge problem should be easier because the base simply moves diagonally along adjacent pegs on the board; there are nine distinct triangles with the same base and height as the original. The second is more challenging because the diagonal does not pass through adjacent pegs. There are exactly two triangles additional triangles with the same base and height as the original, plus a rotation of 180 degrees (not pictured) of the original triangle around the midpoint of the base. Students may find a straight edge beneficial to identify the pegs on the board that are on the line through the apex that is parallel to the base.
The students are not asked to calculate the area of the last two triangles if they are not familiar with the pythagorean theorem. If we continue to take one unit in length as the vertical and horizontal distances between adjacent pegs, the pythagorean theorem is required to calculate the lengths of our base and height.
Using the Pythagorean Theorem to find Area
Using the Pythagorean Theorem to find Area
Using the pythagorean theorem, students can identify that the diagonal length between adjacent pegs is irrational. We can use this to express the length of the base and height in terms of the square root of two and calculate the area. Students can also calculate the length of the base and height directly by forming right triangles with those segments as the hypotenuse of right triangles and using the pythagorean theorem.
If students are familiar with slope, they can also recognize that the slope of the base is 1 and so the apex of any new triangle needs to be on the line of slope 1 through the original vertex. Additionally, the height can be confirmed to have a slope of –1 since perpendicular lines have slopes that are negative reciprocals.
Using the Heron's Formula to find Area
Using the Heron's Formula to find Area
The shears of our last triangle can be verified by ensuring the points lie on a line through the apex that is parallel to the base which has a slope of three-fourths.
Finding the area of this final triangle is more challenging because we don't have a straightforward way of finding the height. Using the pythagorean theorem, however, we can find the lengths of each side. Heron's formula gives a way to calculate the area A of a triangle given side lengths a, b, and c and where s is half the sum of all the side lengths:
For our purposes, an alternative form of Heron's formula, which does not depend on s, will be easier to evaluate because of the two irrational side lengths:
Page updated
Report abuse