# Quod Erat Demonstrandum

## 2011/09/06

### Can you draw…

Filed under: NSS — johnmayhk @ 5:57 上午
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Can you draw a triangle with area of 100 $cm^2$, and two of its sides are 10 cm and 16 cm long? Explain briefly.

（在絕對平面上）理論上，是畫不到的。因為

$100=\frac{1}{2}(10)(16)\sin \theta$

Can you draw a triangle with area of 70 $cm^2$, and two of its sides are 10 cm and 16 cm long? Explain briefly.

No, I can’t. Because I cannot construct $\theta$ which is exactly a solution to $\sin\theta = \frac{7}{8}$.

No, I don’t have a protractor.

## 3 則迴響 »

1. Draw a right-angled triangle with hyp = 4, base = 1 -> height = rt 15
Draw a right angled triangle with base = rt 15, height = 7 -> hyp = 8
Then we can obtain the req. angle.
Ruler is enough to reach this task.

迴響 由 — 2011/09/08 @ 1:18 上午 | 回應

• 楓, Thank you for your reply.

How to construct exactly $90^o$ by using ruler only ?

In theory, it is easy to construct the required $\theta$ by using compass and straightedge (ruler).

Procedure:

1.Draw a line, say L.
2.Mark a point on the line, say A.
3.Draw an circular arc centred at A, intersecting L at B and C (say) by using compass.
4.Draw two circular arcs with same radii, centred at B and C respectively, intersecting at D.
5.Join AD, then AD is perpendicular to L.
6.Fix a unit length.
7.Use compass to copy the unit length, start from A, construct 7 units, mark the point E on L such that AE = 7 units.
8.Use compass to copy the unit length, construct 8 units.
9.Draw a circular arc centred at E with radius 8 units, intersecting AD at F.
10.$\angle AFE$ is the required.

However, I just “say" the procedure (in theory) and do nothing in practice. I really don’t know whether I can draw an angle $\theta$ exactly being an acute angle satisfying $\sin\theta =\frac{7}{8}$ or not, even I’m given tools like paper, ruler, compass, pencil whatever. There may always be errors in human eyes and hands, how can I ensure the construction is error-free? If I’m given a super graphic computer to create the angle, but, the original question is “can you draw…", I can “order" the computer to “draw", however, it is not drawn by me.

If students give the above as a reply to the original question, is it acceptable? Or, should the question be rephrased?

迴響 由 johnmayhk — 2011/09/08 @ 5:09 上午 | 回應

2. i NOTICE THAT the angle is an irrational number similar to Pi. Even you can use the super computer in japan. the computer can’t draw an accurate angle. ^_^

迴響 由 Toi — 2011/12/04 @ 8:43 上午 | 回應