Bruce draws rings of rhombuses about a common centre point. All rhombuses have the same side length.
Rhombuses in the first, or inner, ring are all identical. Each rhombus has a vertex at the centre and each of its sides that meet at the centre is shared with another rhombus. They all have the same size angle at the centre. Figure 1 shows a first ring with 7 rhombuses.
Each rhombus in the second ring has two adjacent sides each of which is shared with a rhombus in the first ring. Figure 2 shows the second ring when the first ring contains 7 rhombuses.
Bruce continues adding rings of rhombuses in the same way for as long as possible. Figure 3 shows the third ring when the first ring contains 7 rhombuses. In this example, since it is not possible to draw any new rhombuses that share an edge with two rhombuses in the third ring, there are only three rings in this rhombus ring pattern.
a) In another rhombus ring pattern, each rhombus in the last ring has an angle of 20 degrees and each rhombus in the second last ring has an angle of 60 degrees. How many rhombuses are in each ring, and how many rings are there in this rhombus ring pattern?
b) A rhombus ring pattern has r rings and the rhombuses in its first ring have central angle** a. What are the angles in a rhombus in th*e *rth ring?
c) A rhombus ring pattern has 7 rings in total. How many rhombuses could there be in each ring?
Figures 1, 2 and 3 can be found at this link.
bruv stop cheating even though im still waiting on the reply to this one.
ok so this is due tomorrow
To answer these questions, we need to analyze the given information about rhombus ring patterns. Let's break down each question and explain how to find the answers.
a) In this rhombus ring pattern:
- The angle of each rhombus in the last ring is 20 degrees.
- The angle of each rhombus in the second-last ring is 60 degrees.
To find the number of rhombuses in each ring, we need to determine the relationship between the angles and the number of rhombuses. In the given pattern, each rhombus in the first ring has an angle of x degrees. The number of rhombuses in the first ring is given as 7.
To find the number of rhombuses in the second ring, we observe that each rhombus in the first ring shares two adjacent sides with rhombuses in the second ring. Given that each angle measures 60 degrees in the second ring, we can calculate the number of rhombuses by dividing 360 degrees (the total angle at the center) by the angle of each rhombus in the second ring. So, 360 / 60 = 6 rhombuses in the second ring.
Similarly, for the third ring, each rhombus shares two adjacent sides with rhombuses in the second ring. Since each angle in the third ring is 20 degrees, the number of rhombuses can be calculated as 360 / 20 = 18 rhombuses in the third ring.
Therefore, in this rhombus ring pattern, the number of rhombuses in each ring is as follows:
- First ring: 7 rhombuses
- Second ring: 6 rhombuses
- Third ring: 18 rhombuses
We can see that the number of rhombuses increases as we move to outer rings in this pattern.
b) In a rhombus ring pattern with r rings, the angles in a rhombus in the r-th ring can be found using the concept of central angles.
In the first ring, each rhombus has a central angle of a degrees. The central angle represents the angle formed at the center of the pattern by the sides of each rhombus.
As we move to the outer rings, each rhombus's central angle will be twice the central angle of the previous ring. This is because each rhombus in the n-th ring shares two adjacent sides with rhombuses in the (n-1)-th ring, resulting in a doubling of the central angle.
So, we can express the angles in a rhombus in the r-th ring as: a, 2a, 4a, 8a, ..., 2^(r-1) * a.
c) In a rhombus ring pattern with 7 rings in total, we need to determine the maximum number of rhombuses that can be in each ring.
To do this, we can observe that each rhombus in the first ring shares two adjacent sides with rhombuses in the second ring. Similarly, each rhombus in the second ring shares two adjacent sides with rhombuses in the third ring, and so on. This pattern continues until the last ring, which cannot share its sides with another ring.
Since we know that the pattern stops at the 7th ring, we can conclude that the 6th and 7th rings will not share their sides with any other ring. Therefore, the maximum number of rhombuses will be in the 6th ring.
To calculate the maximum number of rhombuses in the 6th ring, we can use the concept of central angles. Each rhombus's central angle in the 6th ring will be twice the central angle in the 5th ring. Let's assume the central angle in the 5th ring is x degrees. Therefore, the central angle in the 6th ring will be 2x degrees.
Using the formula mentioned earlier (360 degrees divided by the central angle to find the number of rhombuses), we get 360 / 2x = 180 / x rhombuses in the 6th ring.
Since the number of rhombuses must be a whole number, we need to find a value of x that gives an integer value.
Possible values for x can be multiples of 180 (e.g., 180, 360, 540, ...). However, we need to select the smallest value of x that results in an integer number of rhombuses in the 6th ring.
To find this value, we need to try different multiples of 180 and see which one gives an integer when divided by x.
By trial and error, we can determine that x = 30 degrees satisfies this condition. Therefore, the number of rhombuses in each ring in this rhombus ring pattern would be as follows:
- First ring: Given in the question (not mentioned)
- Second ring: Not calculated
- Third ring: Not calculated
- Fourth ring: Not calculated
- Fifth ring: Not calculated
- Sixth ring: 180 / 30 = 6 rhombuses
Remember to always verify the calculations and assumptions made for each specific problem or pattern given.