Geometry The sides of a square are 16 inches in length. A...

Name: Problem 87, Chapter 8: Precalculus with Limits : A Graphing Approach
Uploaded: 2020-10-12T18:15:30-08:00
Duration: 3 min 39 s
Channel: Joshua Eastwood
Description: Geometry The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the resulting triangles are shaded (see figure on the next page). If this process is repeated five more times, determine the total area of the shaded region.

Geometry The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the resulting triangles are shaded (see figure on the next page). If this process is repeated five more times, determine the total area of the shaded region.

Key Concepts

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero constant called the common ratio. In the context of repeated geometric constructions, the areas of the shapes (or parts of the shapes) form a geometric series. Summing these series is crucial for finding the total area of shaded (or unshaded) regions when the process is repeated multiple times.

Area Partitioning

This concept deals with dividing a geometric shape into parts to analyze the areas of each sub-region. In problems that involve shading portions of a shape, understanding how the total area is partitioned into smaller areas—such as triangles or smaller squares—is key. This aids in calculating the sum of areas of these regions, especially when the partitioning process is repeated iteratively.

Midpoint Square Construction

This concept involves constructing a new square by connecting the midpoints of the sides of an existing square. The resulting figure is also a square, and due to the symmetry of the original, the new square is rotated by 45 degrees relative to the original. This method is used to create nested squares, and it highlights the geometric relationships between the original shape and its derivative figures.

Geometric Similarity and Scaling

When a new square is formed by connecting the midpoints of an existing square, the resulting square is similar to the original, meaning all corresponding angles are equal and the sides are in proportion. The scale factor between the successive squares is constant, which implies that the areas of the squares reduce by a fixed ratio with each iteration. This concept is essential for understanding how repeated geometric constructions impact the size and area of the shapes involved.

Transcript

00:01 This question starts with a square.

00:03 So we'll draw a square.

00:07 And then what it does is it takes the square.

00:09 I'll make it a little bit bigger.

00:12 And it takes the midpoint of each side.

00:18 So here and here.

00:20 And it connects them all the way along.

00:25 Okay.

00:27 And then it shades two of the triangles that are resulting.

00:32 So if the original sides are 16, it tells us.

00:36 16 inches then if we take the midpoint that means this side right here is 8 inches and this side is 8 inches and so the area of that little triangle that results is half the base times the height so it's half of 8 times 8 which is 64 divided by 2 which is 32 square inches okay and because that's an 8 8 triangle these are 45 degrees, which means this is 8 root 2, that result there.

01:10 Okay? so if we shade two of these triangles, the original ones, then our total area so far is going to be 32 plus 32.

01:19 Okay, so we're going to keep that in mind.

01:21 And then what we're going to do is we're going to do this again.

01:25 So we're going to take 8 root 2, take half way along, and draw another square.

01:33 That means that each of these side lengths is 4 root 2, right, half of 8...

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