Chapter 9, Problem 5TP

Solution

To identify: The equality of obtuse angle in two triangles

The two obtuse angles formed by the molecules of liquid and ice form of water are not equal.

Given data: The diagram showing distances between atoms in water molecule in liquid and ice forms is shown in Fig.1. Here, to determine the equality of marked obtuse angles x and y .

  

Method/Formula used:

In a triangle ABC , the angles are determined using Law of Cosines given as:

  $\cos B=\frac{c^2+a^2-b^2}{2ca}$ … (1)

  

Calculations:

In the triangle formed by the molecule of liquid form of water the angle x , using Law of Cosine is calculated as:

Substitute 96 for both a and c , and 165 for b in result (1),

  $\begin{array}{c}\cos x=\frac{{96}^2+{96}^2-{151.8}^2}{2\left(96\right)\left(96\right)}\\ =-0.2502\\ x\Rightarrow {\cos }^{-1}\left(-0.2502\right)\\ x\Rightarrow {104.5}^{\text{ο}}\end{array}$

In the triangle formed by the molecule of ice form of water the angle y , using Law of Cosine is calculated as:

Substitute 101 for both a and c , and 151.8 for b in result (1),

  $\begin{array}{c}\cos y=\frac{{101}^2+{101}^2-{165}^2}{2\left(101\right)\left(101\right)}\\ =-0.3344\\ y\Rightarrow {\cos }^{-1}\left(-0.3344\right)\\ y\Rightarrow {109.5}^{\text{ο}}\end{array}$

Thus, the two obtuse angles formed by the molecules of liquid and ice form of water are not equal.