The sides of a square are 16 cm in length. A new square is formed by joining the midpoints of th

step 1 given a couple of pictures of squares inside squares. And our first job is to figure out if this

The sides of a square are 16 cm in length. A new square is...

The sides of a square are $16 \mathrm{cm}$ in length. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are coloured as shown. a) If the process is repeated 6 more times, determine the total area of the shaded region. b) If the process is repeated indefinitely, find the total area of the shaded region. (GRAPHS CAN'T COPY)

The sides of a square are $16 \mathrm{cm}$ in length. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are coloured as shown.
a) If the process is repeated 6 more times, determine the total area of the shaded region.
b) If the process is repeated indefinitely, find the total area of the shaded region.
(GRAPHS CAN'T COPY)

Key Concepts

Infinite Series Convergence

Infinite series convergence refers to the condition under which the sum of an infinite number of diminishing terms approaches a finite limit. This concept is applicable when the geometric series representing the areas of the iterated squares is extended indefinitely, allowing for the calculation of a total area.

Geometric Series

A geometric series is a sequence where each term is a fixed multiple of the previous one. It is important in this context because the areas of the shapes created in each iteration form a geometric progression, and summing these areas involves the use of geometric series formulas.

Square Geometry

Square geometry involves the properties of squares, including equal side lengths, symmetry, and formulas for area and perimeter. This understanding is fundamental when calculating areas and examining transformations that alter the dimensions of squares.

Midpoint Connection

Connecting the midpoints of a square’s sides creates a new square, typically oriented differently and scaled relative to the original. This construction uses the symmetry of the square and introduces predictable scaling factors essential for analyzing area changes in iterative geometric processes.

Iterative Process

The iterative process in geometry is the repetition of a defined operation, such as forming new squares by joining midpoints. Each iteration results in a new figure whose dimensions and area are consistently scaled, leading to a sequence that can be analyzed mathematically.

Transcript

00:01 Given a couple of pictures of squares inside squares.

00:06 And our first job is to figure out if this process is repeated six more times, what would the total area be? so i've drawn the first two diagrams already, and we're going to try to model a pattern for the areas.

00:21 So the first one, we're being told that it's an original square of 16 by 16.

00:31 So notice that only two of these little triangles are shaded, and we can also think about it in this way.

00:39 If i drew some more hypothetical lines in between, we can see that it forms eight equal triangles, and two of them are shaded.

00:51 So we're really saying one quarter of this area is shaded.

00:58 Now, if one quarter is shaded and we use a highlighter here, this part in the middle, we actually know exactly how much area that is because that's four eights.

01:10 So that's actually half of the original area.

01:14 So in the next picture, you know, we've drawn the square using the midpoints.

01:18 We know that this is already half of the area.

01:22 Of the original.

01:25 And then it turns out, again, by drawing those hypothetical lines, by splitting them up one more time, this actually creates another quarter of the inside square.

01:44 So what we have here is the highlighted part is half of the original area, and then we're taking a quarter of it.

01:55 And it turns out that this pattern is going to hold.

01:58 The next one is going to be one quarter of half of one half a.

02:07 So really one half squared.

02:14 So if we needed to model this to get a general term, a .n, the area of the nth square, is equal to one quarter times one half to the n -minus one...

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