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Asia-Pacific Forum on Science
Learning and Teaching, Volume 5, Issue 2, Article 6 (Aug., 2004) MAN Yiu Kwong Solving geometry problems via mechanical principles
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Asia-Pacific Forum on Science
Learning and Teaching, Volume 5, Issue 2, Article 6 (Aug., 2004) MAN Yiu Kwong Solving geometry problems via mechanical principles
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An area problem of a triangle
The second example is concerned with an area problem of a triangle, as shown in Figure 2. If the area of ΔABC is a cm2 and the points L, M, N divide the line segments BC, CA and AB in the ratio of , respectively, what is the area of ΔPQR?
Figure 2We illustrate how to use the principle of equilibrium to solve this problem. First, let us assume there are three very small solids, with masses 1g, 2g and 4g, located at the points A, B and C respectively. Since AN : NB = 2 : 1, so N is the center of gravity of AB, with a mass of 3g. Similarly, L is the center of gravity of BC, with a mass of 6g. Now, the center of gravity of must lie on both CN and AL, which means at P. For equilibrium, the moment of the masses at A and L about P must be equal, so 1 × AP = 6 × LP. Hence, AP : LP = 6 : 1. Since BL : LC = 2 : 1, so
. Using the same arguments, we have
. Therefore,
.
If this problem is modified to find the ratio of the area of ΔPQR to that of ΔABC, with AN : NB = NL : LC = CM : MA = m : n [1] , we can see that this method works equally well.
[1] The answer will be
.
Copyright (C) 2004 HKIEd APFSLT. Volume 5, Issue 2, Article 6 (Aug., 2004). All Rights Reserved.
