Creating Cross Sections by Dilating
- Let’s create cross sections by doing dilations.
3.1: Dilating, Again
Dilate triangle \(BCD\) using center \(P\) and a scale factor of 2.
Look at your drawing. What do you notice? What do you wonder?
3.2: Pyramid Mobile
Your teacher will give you sheets of paper. Each student in the group should take one sheet of paper and complete these steps:
- Locate and mark the center of your sheet of paper by drawing diagonals or another method.
- Each student should choose one scale factor from the table. On your paper, draw a dilation of the entire sheet of paper, using the center you marked as the center of dilation.
- Measure the length and width of your dilated rectangle and calculate its area. Record the data in the table.
- Cut out your dilated rectangle and make a small hole in the center.
|scale factor, \(k\)||length of scaled rectangle||width of scaled rectangle||area of scaled rectangle|
Now the group as a whole should complete the remaining steps:
- Cut 1 long piece of string (more than 30 centimeters) and 4 shorter pieces of string. Make 4 marks on the long piece of string an equal distance apart.
- Thread the long piece of string through the hole in the largest rectangle. Tie a shorter piece of string beneath it where you made the first mark on the string. This will hold up the rectangle.
- Thread the remaining pieces of paper onto the string from largest to smallest, tying a short piece of string beneath each one at the marks you made.
- Hold up the end of the string to make your cross sections resemble a pyramid. As a group, you may have to steady the cross sections for the pyramid to clearly appear.
Is dilating a square using a factor of 0.9, then dilating the image using scale factor 0.9 the same as dilating the original square using a factor of 0.8? Explain or show your reasoning.
Imagine a triangle lying flat on your desk, and a point \(P\) directly above the triangle. If we dilate the triangle using center \(P\) and scale factor \(k=\frac12\) or 0.5, together the triangles resemble cross sections of a pyramid.
We can add in more cross sections. This image includes two more cross sections, one with scale factor \(k=0.25\) and one with scale factor \(k=0.75\). The triangle with scale factor \(k=1\) is the base of the pyramid, and if we dilate with scale factor \(k=0\) we get a single point at the very top of the pyramid.
Each triangle’s side lengths are a factor of \(k\) times the corresponding side length in the base. For example, for the cross section with \(k=\frac12\), each side length is half the length of the base’s side lengths.
- axis of rotation
A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.
A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.
- cross section
The figure formed by intersecting a solid with a plane.
A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.
Any flat surface on a three-dimensional figure is a face.
A cube has 6 faces.
A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”
A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”
- solid of rotation
A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.
The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.
A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.