Rings and Diamonds
use Pythagoras’ theorem to find the area of a rhombus
use rulers and compasses to make a construction requiring perpendicular bisectors
devise and use problem solving strategies to explore situations mathematically (be systematic, draw diagram)
The construction part of this problem is hopefully a reasonably straightforward application of standard constructions. This might help students to see how to solve the area part as, at first glance, there appears to be insufficient information to solve this.
There are a number of problems of this type. That is, there are a number of problems that appear to have insufficient information. The Pigs, Goats and Sheep problem was just like this. There we were able to use the fact that whole numbers were involved in order to get the extra piece of information that we needed.
Here, as it is a geometrical problem, there is bound to be something in the geometry that will help. But if all else fails, students can go back to their construction and measure the missing side length of the rhombus. This should then help them to see how to obtain it directly.
It’s always worthwhile having a method up your sleeve that may not be exact but which might give you a hint as to how to proceed. Diagrams are often good tools for that sort of thing.
The Problem
The High Peak Jewellery Company wants a new logo. Martin has come up with one based on a ring and a diamond. It is constructed by inscribing a rectangle in a circle as shown below.
The diamond is then drawn inside the rectangle.
Can Martin construct the shape using only ruler and compasses?
What is the area of the diamond shape?
Teaching sequence
- Show the students the logo and ask them to describe it. Encourage them to use geometric terms (bisect, rhombus).
What would you need to construct this?
Where would you start? What next?
- Pose the problem for the students to work on.
- Questions that can help the students get started include:
What information do you know?
What mathematical knowledge could you apply to this problem?
What can you tell me about the radius of the circle?
- As the students work on the problem ask questions that focus on their construction skills and their application of Pythagoras.
What do you need to know to work out the area of the rhombus?
What can you tell me about the side length of the rhombus?
- Ask the students to list the steps they took in solving the problem that they believe are important steps that need to be followed.
- Share lists of steps taken. Discuss the similarities and differences in approaches taken.
Extension
A circle is inscribed in the rhombus. What is its area?
Invent your own logo for The High Peak Jewellery Company.
Solution
To construct the logo Martin has first to draw in the circle. That is straightforward. Now draw in any diameter through the centre of the circle. To get the point which is 3 cm from the centre and the circumference use the normal method of producing the perpendicular bisecting a line segment. Then extend the perpendicular bisector if necessary to cut the circle at two points. Repeat the process on the other side of the centre. Now join the four points on the circle to make the rectangle. Finally join up the points to make the rhombus.
The key to finding the area is to see that the side of the rhombus equals the radius of the circle, since the diagonals of a rectangle are equal. Thus, using Pythagoras’ Theorem, the other side of the triangle is
62 = a2 + 32
27 = a2
√27 = a
3√3 = 27
Hence the area is half the base times the height = 0.5 x 3 x 3√3. This is approximately 7.79 cm2.
Extension:
To find the radius of the circle inscribed in the rhombus, note that the radius of this circle is perpendicular to the side of the rhombus.
Now Sin A = 3/6 using the larger triangle and Sin A = R / (3√3).
Therefore R / (3√3) = 3/6
Giving R = (9√3)/ 6 or 2.6.
| Attachment | Size |
|---|---|
| RingsandDiamonds.pdf | 59.01 KB |
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