A square is formed by joining the midpoints of alternate sides of a regular octagon with side length 14 in. Find the shaded area
.
1 answer:
Answer:
98(1 + 2√2) in² ≈ 375 in²
Step-by-step explanation:
Assuming the shaded region is outside of the square and inside of the octagon, we can find the area by subtracting the area of the square from the area of the octagon.
The area of a regular octagon is 2 (1 + √2) s². We can show this by finding the area of the square outside of the octagon, and subtracting the triangles in the corners:
(s + √2 s)² − 4 (½ (½√2 s)²)
(1 + √2)² s² − 4 (½ (s²/2))
(1 + 2√2 + 2) s² − s²
2 (1 +√2) s²
The diagonal of the inner square is equal to the width of the octagon, (1+√2) s. So the side length of the square is:
½√2 (1+√2) s
½(2+√2) s
The area of the square is therefore:
(½(2+√2) s)²
¼(2+√2)² s²
¼(4+4√2+2) s²
¼(6+4√2) s²
½(3+2√2) s²
The area of the shaded region is therefore:
2 (1 +√2) s² − ½(3+2√2) s²
½ s² (4 (1 +√2) − (3+2√2))
½ s² (4 + 4√2 − 3 − 2√2)
½ s² (1 + 2√2)
The side length of the octagon is s = 14 in, so the area is:
½ (14 in)² (1 + 2√2)
= 98(1 + 2√2) in²
≈ 375 in²
You might be interested in
Answer:
23.1% probability of meeting at least one person with the flu
Step-by-step explanation:
For each encounter, there are only two possible outcomes. Either the person has the flu, or the person does not. The probability of a person having the flu is independent of any other person. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
In which is the number of different combinations of x objects from a set of n elements, given by the following formula.
And p is the probability of X happening.
Infection rate of 2%
This means that
Thirteen random encounters
This means that
Probability of meeting at least one person with the flu
Either you meet none, or you meet at least one. The sum of the probabilities of these outcomes is 1. So
We want . Then
In which
23.1% probability of meeting at least one person with the flu
<h3>Answer:</h3>
Any 1 of the following transformations will work. There are others that are also possible.
- translation up 4 units, followed by rotation CCW by 90°.
- rotation CCW by 90°, followed by translation left 4 units.
- rotation CCW 90° about the center (-2, -2).
The order of vertices ABC is clockwise, as is the order of vertices A'B'C'. Thus, if reflection is involved, there are two (or some other even number of) reflections.
The orientation of line CA is to the east. The orientation of line C'A' is to the north, so the figure has been rotated 90° CCW. In general, such rotation can be accomplished by a single transformation about a suitably chosen center. Here, we're told there is <em>a sequence of transformations</em> involved, so a single rotation is probably not of interest.
If we rotate the figure 90° CCW, we find it ends up 4 units east of the final position. So, one possible transformation is 90° CCW + translation left 4 units.
If we rotate the final figure 90° CW, we find it ends up 4 units north of the starting position. So, another possible transformation is translation up 4 units + rotation 90° CCW.
Of course, rotation 90° CCW in either case is the same as rotation 270° CW.
_____
We have described transformations that will work. What we don't know is what is in your drop-down menu lists. There are many other transformations that will also work, so guessing the one you have available is difficult.