6.3 Geometry  (Page 3/7)

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We use the symbol $\equiv$ to mean is similar to .

Similar Polygons

Two polygons are similar if:

1. their corresponding angles are equal, and
2. the ratios of corresponding sides are equal.

Show that the following two polygons are similar.

1. We are required to show that the pair of polygons is similar. We can do this by showing that the ratio of corresponding sides is equal and by showing that corresponding angles are equal.

2. We are given the angles. So, we can show that corresponding angles are equal.

3. All angles are given to be 90 ${}^{\circ }$ and

$\begin{array}{ccc}\hfill \stackrel{^}{A}& =& \stackrel{^}{E}\hfill \\ \hfill \stackrel{^}{B}& =& \stackrel{^}{F}\hfill \\ \hfill \stackrel{^}{C}& =& \stackrel{^}{G}\hfill \\ \hfill \stackrel{^}{D}& =& \stackrel{^}{H}\hfill \end{array}$
4. We first need to see which sides correspond. The rectangles have two equal long sides and two equal short sides. We need to compare the ratio of the long side lengths of the two different rectangles as well as the ratio of the short side lenghts.

Long sides, large rectangle values over small rectangle values:

$\begin{array}{ccc}\hfill \mathrm{Ratio}& =& \frac{2L}{L}\hfill \\ & =& 2\hfill \end{array}$

Short sides, large rectangle values over small rectangle values:

$\begin{array}{ccc}\hfill \mathrm{Ratio}& =& \frac{L}{\frac{1}{2}L}\hfill \\ & =& \frac{1}{\frac{1}{2}}\hfill \\ & =& 2\hfill \end{array}$

The ratios of the corresponding sides are equal, 2 in this case.

5. Since corresponding angles are equal and the ratios of the corresponding sides are equal the polygons ABCD and EFGH are similar.

All squares are similar.

If two pentagons ABCDE and GHJKL are similar, determine the lengths of the sides and angles labelled with letters:

1. We are given that ABCDE and GHJKL are similar. This means that:

$\frac{\mathrm{AB}}{\mathrm{GH}}=\frac{\mathrm{BC}}{\mathrm{HJ}}=\frac{\mathrm{CD}}{\mathrm{JK}}=\frac{\mathrm{DE}}{\mathrm{KL}}=\frac{\mathrm{EA}}{\mathrm{LG}}$

and

$\begin{array}{ccc}\hfill \stackrel{^}{A}& =& \stackrel{^}{G}\hfill \\ \hfill \stackrel{^}{B}& =& \stackrel{^}{H}\hfill \\ \hfill \stackrel{^}{C}& =& \stackrel{^}{J}\hfill \\ \hfill \stackrel{^}{D}& =& \stackrel{^}{K}\hfill \\ \hfill \stackrel{^}{E}& =& \stackrel{^}{L}\hfill \end{array}$
2. We are required to determine the

1. $a$ , $b$ , $c$ and $d$ , and
2. $e$ , $f$ and $g$ .
3. The corresponding angles are equal, so no calculation is needed. We are given one pair of sides $DC$ and $KJ$ that correspond. $\frac{DC}{KJ}=\frac{4,5}{3}=1,5$ so we know that all sides of $KJHGL$ are 1,5 times smaller than $ABCDE$ .

4. $\begin{array}{ccc}\hfill \frac{a}{2}=1,5& \therefore & a=2×1,5=3\hfill \\ \hfill \frac{b}{1,5}=1,5& \therefore & b=1,5×1,5=2,25\hfill \\ \hfill \frac{6}{c}=1,5& \therefore & c=6÷1,5=4\hfill \\ \hfill d=\frac{3}{1,5}& \therefore & d=2\hfill \end{array}$
5. $\begin{array}{ccc}\hfill e& =& {92}^{\circ }\left(\mathsf{corresponds}\mathsf{to}\mathsf{H}\right)\hfill \\ \hfill f& =& {120}^{\circ }\left(\mathsf{corresponds}\mathsf{to}\mathsf{D}\right)\hfill \\ \hfill g& =& {40}^{\circ }\left(\mathsf{corresponds}\mathsf{to}\mathsf{E}\right)\hfill \end{array}$
6. $\begin{array}{ccc}\hfill a& =& 3\hfill \\ \hfill b& =& 2,25\hfill \\ \hfill c& =& 4\hfill \\ \hfill d& =& 2\hfill \\ \hfill e& =& {92}^{\circ }\hfill \\ \hfill f& =& {120}^{\circ }\hfill \\ \hfill g& =& {40}^{\circ }\hfill \end{array}$

Similarity of equilateral triangles

Working in pairs, show that all equilateral triangles are similar.

Polygons-mixed

1. Find the values of the unknowns in each case. Give reasons.
2. Find the angles and lengths marked with letters in the following figures:

Introduction

Analytical geometry, also called co-ordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra, and the Cartesian co-ordinate system. It is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.

Distance between two points

One of the simplest things that can be done with analytical geometry is to calculate the distance between two points. Distance is a number that describes how far apart two point are. For example, point $P$ has co-ordinates $\left(2,1\right)$ and point $Q$ has co-ordinates $\left(-2,-2\right)$ . How far apart are points $P$ and $Q$ ? In the figure, this means how long is the dashed line?

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