A mosaic is designed in the shape of an equilateral...

Name: Problem 63, Chapter 9: College Algebra
Uploaded: 2020-05-09T20:35:09-08:00
Duration: 3 min 10 s
Channel: Julie Silva
Description: A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the illustration. How many tiles of each color will be required? CANT COPY THE IMAGE

A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the illustration. How many tiles of each color will be required? CANT COPY THE IMAGE

Key Concepts

Alternating Patterns in Tiling

Alternating patterns refer to arrangements where different types or colors of tiles are placed in a regular sequence such that no two adjacent tiles share the same characteristic. This concept is applied in mosaic designs to create visual patterns and can involve combinatorial reasoning to determine the number of tiles needed for each color.

Tessellation and Geometric Dissection

Tessellation involves covering a plane or a larger shape completely using copies of a smaller shape without gaps or overlaps. In the context of equilateral triangles, the larger triangle can be subdivided into a series of smaller triangles by drawing lines parallel to its sides, a method that is essential in solving mosaic tiling problems.

Properties of Equilateral Triangles

An equilateral triangle is a triangle in which all three sides are of equal length and all three interior angles measure 60 degrees. This symmetry allows for the use of specific formulas to calculate area and height, making it a common shape in geometric constructions and tiling problems.

Area Scaling in Similar Figures

When dealing with similar figures, such as triangles with proportional side lengths, the ratio of their areas is equal to the square of the ratio of their sides. This principle is crucial for understanding how a larger geometric shape can be partitioned into smaller congruent shapes.

Transcript

00:01 So as you can see, we have a triangle here that's equilateral, that's 20 feet on each side.

00:06 And it says that each tile in the mosaic is also in this shape of the equilateral triangle, that's 12 inches to a side.

00:13 And the tiles are going to alternate in color shown in the picture.

00:17 So we want to figure out how many tiles of each color are going to be required.

00:20 So let's start with the beige ones.

00:24 So without obviously drawing out this whole thing, and you obviously certainly don't want to count up every single base triangle in this picture.

00:31 Picture, but let's go through each row and count up for the first couple rows anyways, how many triangles will be there? well, in the first row, where there's only one triangle, and it's beige, so there's one base triangle in the first row.

00:45 Well, in the second row, there's two base triangles.

00:49 If you look at the third row, there's three base triangles.

00:53 In the fourth row, there's four base triangles.

00:55 So if you notice in each row, it continues to go up by 1.

00:59 Well, let's think about it.

01:01 If this is 20 feet and each of those equilateral triangles has a length of 20, well, in this particular case, there's going to be 20 rows.

01:09 So that would mean that in the last row there would be 20 tiles.

01:14 So we just have to find the sum of this series.

01:16 Well, think about what we have.

01:18 We know that a sub 1 is equal to 1.

01:20 We know that a sub n is equal to 20.

01:23 And like we mentioned, there are 20 rows.