Lauretta J. Fox
When a new building is being designed, the architect must convert his ideas to drawings. These drawings enable homeowners, contractors, carpenters, and others to know exactly what the architect has in mind. They show the sizes, shapes and arrangements of rooms, structural parts, windows, doors, closets and other important details of construction. The pictures are miniature reproductions of the building and are called
scale
drawings
.
Scale drawings which represent the parts of a building must be in exact proportion to the actual structure. Various scales may be used for this purpose. For example, 1/8 inch may be used to represent one foot, that is, instead of drawing an object one foot long, one would draw it 1/8 inch long. One of the most common scales used by architects is 1/4 inch = one foot.
Example 1:

Using the scale 1/8 inch = 1 foot, complete the following:


a. 12 feet are represented by
inches.


b. 1 7/8 inches represent
feet.

Solution
:

a. foot= 1/8 inch



12 feet = 12 x 1/8 = 1 1/2 inches.



12 feet are represented by 1 1/2 inches.


b. 1/8 inch = 1 foot



1 7/8 $dv 1/8 = 1 7/8 x 8 = 15 feet.



1 7/8 inches represent 15 feet.

Example
2
:

Using the scale 1/8 inch = 1 foot, draw a line segment to represent 24 feet.

Solution
:

1 foot = 1/8 inch.


24 feet = 24 x 1/8 = 3 inches.


The scale drawing that represents 24 feet must be 3 inches long.


3 “ = 24 feet

Example 3:

Measure the length and width of rectangle ABCD. Using the scale 1/4 inch = 1 foot, express in feet the length and width of the actual figure represented.

(figure available in print form)
Solution
:

The length of the rectangle is 2 inches and its width is 1/2 inch. The dimensions of the actual figure are obtained as follows:


1 foot = 1/4 inch


Length

Width


1/4 inch = 1 foot

1/4 inch = 1 foot


2 $dv 1/4 = 2x4 = 8 feet

1/2 $dv x 4= 2 feet


Actual Length= 8 feet

Actual Width = 2 feet

Exercises:

1. Using the scale 1/4 inch = 1 foot, find the actual length in feet represented by the following lengths on the drawing: (a) 3 in. (b) 2 1/4 in. (c) 4 3/4 in.

2. Using the scale 1/8 inch = 1 foot, how long a segment should be drawn to represent an object whose actual length is (a) 32 ft. (b) 5 yds. (c) 12 ft. (d) 4 ft.?

3. Make a scale drawing of a rectangular shaped room whose dimensions are 14 feet by 24 feet.

4. Make a scale drawing of the floor plan shown, using the scale 1/4 inch = 1 foot.
(figure available in print form)

5. Measure the length and width of your classroom. Choose a convenient scale and make a scale drawing of the floor plan.

6. Using the scale 1/4 inch = 1 foot, find the actual dimensions of each room in the diagram on the next page.
Suggested
Assignment
:
Measure the rooms in your home. Using the scale 1/4 inch= 1 foot, make a drawing showing the shapes and arrangement of the rooms on each level. Indicate windows by the symbol , fireplaces by the symbol , and doors by a break in the line .
(figure available in print form)
Ratio
From earliest times the Greeks and Romans were preoccupied with building structures that were pleasing to the eye. They were convinced that architectural beauty was obtained by the interrelation of universally valid ratios. Frequently complicated mathematical ratios were used by architects to accomplish their goals.
A
ratio
is a comparison by division of two quantities expressed in the same unit of measure. The ratio may be expressed in words or in symbols. For example, if segment AB is 1 inch long and segment CD is 2 inches long, we say that the ratio of AB to CD is 1 to 2. In symbols, the ratio may be expressed as the fraction 1/2, or it may be written in the form 1:2. The fraction line and the symbol : are taken from the division sign $dv.
Example 1:

The length and width of a room are 22 feet and 14 feet, respectively. Express in three different ways the ratio of the length of the room to the width in simplest form.

Solution
:

(1) 22 to 14 or 11 to 7.


(2) 22/14 or 11/7


(3) 22:14 or 11:7

Example
2
:

A door is 30 inches wide and 2 3/4 yards high. What is the ratio of the width to the height of the door?

Solution
:

Width = 30 inches


Height=2 3/4 yds. = 2 3/4 x 36=11/4 x 36=99 in.


The ratio of the width to the height is 30 to 99 or 10 to 33.

Exercises
:

1. Using a ruler, measure line segments AB, BC, CD, AC, BD, and AD. Evaluate the following ratios:
a.) AB:BC

c.) AB:BD

e.)BD:BC

g.) CD:AB

b.) BC:CD

d.) BC:AD

f.)AC:CD

h.) AD:CD


2. Express each of the following ratios in lowest terms:
a.) 30:35

c.) 4: 1/2

e.).08:.3

b.) 40: 280

d.) 6:.2

f.) 1/5: 7/15


3. Find the ratio of the first quantity to the second:
a.) 3 ft. to 6 yd.

c.) 4.5 in. to 3 1/4 yd.

b.) 8 in. to 5 ft.

d.) 1/2 ft. to 54 in.


4. In the adjacent figure, find the following measures:
Width of wall_______

Height of door _______

Height of wall
_______

Width of window_______

Width of door_______

Height of window_______


Using these dimensions, write all possible ratios.

(figure available in print form)

5. Measure the length, width and height of your classroom. What is the ratio of the (a) length to the width? (b) length to the height? (c) width to the height?
Suggested
Assignment
:
Measure the length, width and height of one room in your house. Find the dimensions of all doors and windows in that room. Using these measurements, write all possible ratios.
Proportion
From earliest times men have recognized the value of good proportions in architecture. The ancient Greeks and Romans followed certain mathematical ratios and proportions to attain order, unity and beauty in their buildings. Using fixed mathematical formulas they were able to establish a pleasing relationship among various parts of buildings that have been admired for generations.
A
proportion
is an equation stating that two ratios are equal. Every proportion has four
terms
. The first and fourth terms are the
extremes
. The second and third terms are the
means
. In every proportion the product of the means equals the product of the extremes. The first equation below shows the order in which the terms of a proportion are written. Equations 2 and 3 demonstrate two ways in which a proportion may be written.
(figure available in print form)
The proportion is read: “2 is to 3 as 6 is to 9”. The means are 3 and 6. The extremes are 2 and 8.
Every proportion may be written in four equivalent forms.
1.) Given

2.) Invert the

3.) Alternate

4.) Invert the


terms in 1.

the terms

terms in 3.



in 2.

(figure available in print form)
A new proportion may also be obtained from a given one by adding or subtracting 1 on both sides of the equation in the following manner:
(figure available in print form)
The fourth term of a proportion is called the
fourth
proportional
to the other three terms. In 1/2 = 3/6, 6 is the fourth proportional to 1, 2, and 3. When the second and third terms of a proportion are the same, they are called the
geometric
mean
or
mean
proportional
, and the fourth term is then called the
third
proportional
. 1/2 = 2/4, 2 is the mean proportional, and 4 is the third proportional.
Example
1
:

Is 2/3 = 5/7 a true proportion?

Solution
:

2/3= 5/7 3(5) = 15 2(7)= 14 3(5)= 2(7) Since the product of the means does not equal the product of the extremes, 2/3 = 5/7 is not a proportion.

Example
2
:

Find the missing term. 4/7 = x/35

Solution
:

4/7= x/35 7x = 4(35) 7x = 140 x = 20 The missing term is 20.

Example
3
:

Find the fourth proportional to 1, 2 and 3.

Solution
:

1/2= 3/x 1x= 2(3) x= 6


The fourth proportional is 6.

Example
4
:

Find the mean proportional between 2 and 8.

Solution:

2/x= x/8 x2= 16 x= 4


The mean proportional between 2 and 8 is 4.

Example
5
:

Given the proportion 3/5 = 8/15, write a new proportion by addition.

Solution
:

3/5 = 8/15 3/5+ 1= 8/15 + 1 8/5= 24/15

Exercises

1. Tell whether each pair of ratios given forms a proportion.

(figure available in print form)

2. Find the missing term.

(figure available in print form)

3. Find the fourth proportional to:

a.) 4, 5, 6 b.) 8, 10, 12 c.) 3, 5, 7 d.) 8, 12, 13

4. Find the mean proportional between:

a.) 3 and 27 b.) 4 and 16 c.) 6 and 24 d.) 2 and 50.

5. Write two new proportions using addition and subtraction. 4/7 = 24/42.
Symmetry
One way to attain balance and repose in an architectural structure is by the use of symmetry.
Symmetry
is a distancepreserving transformation of any figure by reflection, translation or rotation. Two points are symmetric about a line if the line is the perpendicular bisector of the segment joining the two points. In figure 1 points A and B are symmetric about line n, since n is the perpendicular bisector of segment AB. B is said to be the mirror image or reflection of A, and n is the
line
of
symmetry
.
Figure 1: A.
(figure available in print form)
In figure 2, triangle ABC and triangle DEF are symmetric about line m. The corresponding sides and corresponding angles of the triangles are congruent, and m is the perpendicular bisector of segments BE, CF and AD. Triangle DEF is the reflection of triangle ABC, and m is the line of symmetry.
Figure 2: C
(figure available in print form)
Points A and B are symmetric about point M if M is the midpoint of segment AB
(figure available in print form)
Symmetry may be attained by a second distance preserving transformation known as a
translation
. A translation is the composite of two reflections over two parallel lines. In a translation a figure and its image are parallel. In the figure below line p is parallel to line m. Triangle ABC is reflected over line p, then its image triangle DEF is reflected over line m producing triangle GHI congruent to triangle ABC. The same translation may be performed by sliding triangle ABC along the plane to a new position so that the original figure and its image are parallel and congruent.
(figure available in print form)
The third method of attaining symmetry is by
rotation
. A rotation is the composite of two reflections over two intersecting lines. In the figure below, triangle MNP is the reflection of triangle ACB over line p. Triangle GJK is the reflection of triangle MNP over line m. The same image may be obtained by rotating triangle ACB counterclockwise. Triangle GJK is congruent to triangle ACB.
(figure available in print form)
Exercises

1. points A and B have symmetry with respect to the line y= 2. Find point B if point A is (a) (1,3) (b) (0,0) (c) (4,5)

2.) Points C and D have symmetry with respect to the point (3,1). Find D if C is (a) (5,4) (b) (1,2) (c) (0,6).

3. Name the lines of symmetry in the figure shown. Is there a point of symmetry in the figure?

(figure available in print form)

4. Tell whether the following is a reflection, translation or rotation:

(figure available in print form)

5. Which of the following figures illustrate symmetry?
(figure available in print form)
Suggested
Assignment
:
Write a report on architecture of ancient Greece. Sketch a picture of the Parthenon and discuss the proportion and symmetry of its architecture. In the discussion include an explanation of the Golden Ratio.