# Checking the mathematical consistency of geometric figures.

In many countries, teachers often have to set their own questions for tests and examinations: some of them even set their own questions for assignments for students. These teachers do not usually select questions from textbooks used by the students because the latter would have seen the questions. If the teachers take the questions from other sources such as assessment books not used by the school students, this will constitute as plagiarism. Although they can refer to these other sources, they still have to modify the questions. It is during modification or setting their own questions that teachers may encounter problems which they may not be even aware of. For example, in setting tests or assignments on geometry or mensuration, maths teachers often have to design and draw geometric figures (e.g. see Figure 1).The first question is whether it is possible to construct such a geometric figure. For example, Figure 1 is mathematically inconsistent because ACB = 90[degrees] (right angle in semi-circle) since AB is the diameter, but the lengths of the three sides of ABC do not satisfy Pythagoras' Theorem (since [5.sup.2] + [12.sup.2] = [13.sup.2], and not [14.sup.2]). The second question is why is this important? So what if the figure cannot be constructed due to contradictory information? The main issue is that students will get different answers if they use different methods to solve. For example, in Figure 1, if they use [angle]ABC = 90[degrees] (right angle in semi-circle), then the area of [DELTA]ABC will be 2 x base x height = 30 [cm.sup.2]. But if they use cosine rule to calculate [angle]ABC = 103.0[degrees] (to 4 s.f.), then the area of [DELTA]ABC will be 1/2 x 5 x 12 x sin 103.0[degrees]=29.2[cm.sup.2] (to 3 s.f.). Therefore, it is very important for a geometric figure to be mathematically consistent.

Figure 1 is a common mistake that some teachers make, but they may not even be aware of it. Why? One reason is that they may not know that they have to check for mathematical consistency. Another reason is that they may not know how and what to check. The problem with Figure 1 may be obvious to many teachers, but what about the geometric figure shown in Figure 2? Is it mathematically consistent? How do we know how and what to check? (In fact, Figure 2 is mathematically inconsistent, which I will discuss later in this article.)

The failure to check whether a geometric figure is mathematically consistent is not just a problem for school teachers when setting their own questions or choosing a problem from a textbook or from the Internet. It is also a problem for writers of textbooks and assessment books, most of whom are either teachers or former teachers. In this article, I will describe a general method which teachers or writers can use to check whether a given geometric figure, or one that they have designed, is mathematically consistent.

Theoretical or mental construction

We will apply this method to a simpler geometric figure (see Figure 3) first.

First, we will construct Figure 3 using as few pieces of given information as possible. There are two ways to do this. One method is to use mathematical instruments, such as a pair of compasses, protractors and set squares, to construct the actual figure; or to use geometrical software such as GeoGebra or Geometer's Sketchpad (GSP) to do the same. The other easier method is what I call 'theoretical construction', which means we can construct the figure theoretically using mathematical instruments or geometrical software, but we do not actually use the instruments or the software. For example, we can construct Figure 3 theoretically by drawing the line AB using freehand, and labelling 14 cm on it (see Figure 4) because we know we can construct such a line by using a ruler. Then we can draw the line AE using freehand such that it is perpendicular to AB because we know we can construct such a line by using a protractor to measure 90o or by using a set square. Similarly, we can construct the lines ED and BC by using some mathematical instruments, so we just draw the lines using freehand. In fact, all the above theoretical constructions can be done mentally, which is even faster than freehand drawing or the use of mathematical instruments or geometrical software to construct.

The crucial part comes when we realise that the points C and D are fixed, so when we draw the line CD, its length is also fixed. But we have yet to use the given length of 6 cm for CD. This means that we have to check whether the length of CD is really 6 cm. If we have constructed the figure using mathematical instruments or geometrical software, we can just measure the length of CD. Since this is a theoretical construction, we will have to use another method. In this case, we can use Pythagoras' Theorem to find the length of CD easily (see Figure 5), which is actually 5 cm, and not the given 6 cm. In other words, the figure is mathematically inconsistent. The implication for teachers setting this kind of questions is that they cannot arbitrarily assign any length to CD, but they must find the actual length of CD first.

In fact, we can apply the same 'theoretical construction' method for Figure 1. Since the diameter of the circle is 14 cm, then its radius is 7 cm, so we can use a pair of compasses to construct a circle of radius 7 cm (see Figure 6). Then we can draw the diameter AB. Next we can construct the line AC by using a pair of compasses to draw an arc of radius 5 cm to cut the circle at C. The crucial part comes when we realise that the points B and C are fixed, so when we draw the line BC, its length is also fixed. But we have yet to use the given length of 12 cm for BC. This means that we have to check whether the length of BC is really 12 cm by using Pythagoras' theorem since ACB = 90. The implication for teachers setting this kind of questions is that they cannot arbitrarily assign any length to BC, but they must find the actual length of BC first.

However, in this example, there is another way to draw the line AC, where C is below the diameter (see Figure 7). It does not matter because the length of BC will still be the same in both Figure 1 and Figure 7 since one figure is a reflection of the other figure.

To summarise this method, we construct the geometric figure using as few pieces of given information as possible. If we finish constructing a unique figure without using all the given information, then we have to check the unused information to ensure that they are correct. A figure is considered unique regardless of its orientation (i.e., you can reflect or rotate it, and it is still the same figure). For a geometric figure involving angles only (e.g., Q1 in Figure 10 later), it is considered unique regardless of its actual size (i.e., it does not matter how big or small you draw it) because it will not affect the size of any other angle in the figure. In addition, for angles in the same segment, it does not really matter which points on the circumference of the circle the angles are at. There are two methods to construct the figure: either we use mathematical instruments or geometrical software to construct the actual figure (which is troublesome), or we construct the figure theoretically using freehand or mentally (which is faster).

Applying the general method to a complicated example

Let us return to Figure 2. First, we construct a rectangle 15 cm by 6 cm with its diagonal SQ (see Figure 8). Since C is a rectangle, Triangle A (i.e., ASWX) is a right-angled triangle. Given that the area of ASWX is 4 [cm.sup.2], how do we determine the length of SW to construct the line XW? If we use mathematical instruments or geometrical software to construct, we have to use similar triangles to calculate the length of SW first, which is more troublesome. The beauty of theoretical construction is that we do not even need to know the length of SW. All we need to know is whether the length of SW is fixed. This can be achieved by using the following argument. Since [DELTA]SWX = 90[degrees], there is only one way to draw XW such that the area of ASWX is 4 [cm.sup.2], because if you draw any other lines such as [X.sub.1][W.sub.1] or [X.sub.2][W.sub.2], then the area of [DELTA]SWX will either be less than or more than 4 [cm.sup.2] respectively. (If [DELTA]SWX is not 90[degrees], then there are many ways to draw XW since XW does not have to be vertical.) Since there is only one way to draw XW in this example, then the length of SW is fixed.

Once SW and XW are fixed, there is only one way to draw VX since VXW is a straight line (see Figure 9). This means that the area of Triangle B is fixed. If the setter wants to give this extra information, he or she cannot arbitrarily assign a value to the area of Triangle B. Instead, the setter can make use of similar triangles to determine the area of Triangle B, which is about 22.2 [cm.sup.2], not 16 [cm.sup.2] as stated in the question. (If the figure is mathematically consistent and there is no need to give a value to the area of Triangle B, then we can skip the tedious process of using similar triangles altogether. But if we use mathematical instruments or geometrical software to construct, we have to use similar triangles right at the start to find the length of SW first, which is more troublesome.)

If the students make use of the given area of Triangle B, i.e. 16 [cm.sup.2], then the answer for the area of Rectangle C will be 45-4-16 = 25 [cm.sup.2]. If the students ignore the given area of Triangle B and make use of similar triangles, then the answer for the area of Rectangle C will be 45-4-22.2 = 18.8 [cm.sup.2]. Therefore, it is very important for a geometric figure to be mathematically consistent.

Moreover, 22.2 [cm.sup.2] is not a good value to use, not because it is not a whole number, but because it is an approximate value. Nevertheless, if we want to make all the values into whole numbers, we will have to change the area of Triangle A. The only possibilities for XW are 1 cm, 2 cm, 3 cm, 4 cm and 5 cm because QR = 6 cm. Since [DELTA]SWX is similar to ASRQ, SW/XW = SR/RQ = 2.5. Then we can find the length of SW and thus the area of Triangle A (see Table 1). Similarly, since [DELTA]QVX is similar to [DELTA]SWX, VQ/RX = SW/XW = 2.5, and we can find the area of Triangle B.

From Table 1, we see that the only whole numbers for the areas of Triangle A and Triangle B are 5 [cm.sup.2] and 20 [cm.sup.2] respectively, and vice versa. In fact, XW must be even for SW to be a whole number since SW/XW = 2.5 , i.e., SW = 5/2 XW; so we can actually reject all the odd values of XW right from the start, i.e., start with XW = 2 cm or 4 cm, and then check whether the areas of the two triangles are whole numbers.

Another way to find whole numbers for the areas of Triangle A and Triangle B is as follow. Let SW = x. Then SW/XW = 2.5, i.e., XW = 5/2 x; and so the area of Triangle A is 1/2 x (2/5 x) = [x.sup.2]/5. Thus, for the area of Triangle A to be a whole number, x=5 or 10 (since 0< x <15), i.e., area of Triangle A = 5 [cm.sup.2] or 20 [cm.sup.2].

Therefore, if we rephrase the question such that the area of Triangle A is 5 [cm.sup.2] and the area of Triangle B is 20 [cm.sup.2], then the question will not only be mathematically consistent, but it will also contain all whole numbers. In fact, the answer for the area of Rectangle C will also be a whole number, namely 20 [cm.sup.2].

Conclusion

In this article, I have described a general method, called theoretical or mental construction, which teachers or writers can use to check whether a given geometric figure, or one that they have designed, is mathematically consistent. In addition, for some examples, I have also discussed how to find the correct value without constructing the actual figure, or how to adjust the values to more suitable ones, such as whole numbers.

It is beyond the scope of this article to discuss another type of issue with geometric figures: a given figure does not contain enough information to solve the problem, i.e. the figure is ambiguous. Nevertheless, the reader can apply the same construction method to decide whether a geometric figure is ambiguous: if there is not enough given information to construct the figure uniquely, then the figure is ambiguous.

Exercise

I will leave the reader with some geometric figures (see Figure 10) to determine whether they are mathematically consistent. If you can construct a unique figure theoretically or mentally without using all the given information, then you have to check the unused information to ensure that they are correct. A figure is considered unique regardless of its orientation; and for a figure involving angles only (e.g., 01), it is considered unique regardless of its size. In addition, for angles in the same segment, it does not really matter which points on the circumference of the circle the angles are at. Sometimes, a question does not draw any geometric figures, but it may actually involve them (e.g., 03).

Joseph Yeo

National Institute of Education, Singapore

<josephbw.yeo@nie.edu.sg>

Caption: Figure 1. Is this geometric figure mathematically consistent?

Caption: Figure 2. How do we check whether this geometric figure is mathematically consistent?

Caption: Figure 3. How do we check whether this geometric figure is mathematically consistent?

Caption: Figure 4. Constructing Figure 3 theoretically by drawing freehand.

Caption: Figure 5. Checking the length of BC in Figure 3.

Caption: Figure 6. Constructing Figure 1 theoretically by drawing freehand.

Caption: Figure 7. Another way to construct Figure 1.

Caption: Figure 8. Constructing Figure 2 theoretically.

Caption: Figure 9. Constructing Figure 2 theoretically (continued).

Caption: Figure 10. How do we check whether these geometric figures are mathematically consistent?

Table 1. Possible values for areas of Triangles A and B. Area of [DELTA]A/ Area of AB/ XW/cm SW/cm [cm.sup.2] VX/cm VQ/cm [cm.sup.2] 1 2.5 1.25 5 12.5 31.25 2 5 5 4 10 20 3 7.5 11.25 3 7.5 11.25 4 10 20 2 5 5 5 12.5 31.25 1 2.5 1.25

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Title Annotation: | mathematical consistency |
---|---|

Author: | Yeo, Joseph |

Publication: | Australian Mathematics Teacher |

Article Type: | Essay |

Geographic Code: | 8AUST |

Date: | Dec 22, 2017 |

Words: | 2618 |

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