Chapter 6
Similarity and Proportion
A photo can fill a small screen or a large poster and still look like the same picture. Yet dragging just one edge can make every face look wrong. What separates a change in size from a change in shape? Similarity gives us a way to ask that question precisely. Two figures can be different sizes, but what would it take to be sure that their shapes really match?
What You'll Explore
6 lessons.
Progress can't be saved in this browser, so lesson statuses can't be set here.
-
Imagine enlarging a triangle on a grid while keeping its shape familiar. You'll explore how choosing a center and a scale factor controls the new figure, and what to examine when a change in size starts to look like a stretch.
-
Can triangles that point in different directions still have the same shape? You'll investigate what it means for triangles to be similar and how much their angle measurements can tell you, even when their side lengths differ.
-
Suppose you know several side lengths but only a little about the angles of two triangles. You'll investigate when those measurements provide enough evidence to decide whether the triangles have the same shape.
-
Draw a segment joining two sides of a triangle, parallel to the third side. What relationships might connect the lengths in the picture, especially when the new segment joins the midpoints of those sides?
-
Splitting one angle of a triangle into equal parts sends a segment toward the opposite side. You'll investigate where it lands and what connection there might be between that location and the triangle's other side lengths.
-
A tree's height may be difficult to measure directly, even when its shadow is easy to reach. You'll explore when similarity can connect measurements you can make to distances you cannot reach, and what makes such a geometric model trustworthy.