The theorem of Pythagoras says that 'In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides'. When this theorem is shown pictorially. it Is usuatly illustrated by a triangle with squares drawn on the sides. This tends to be a little misleading since the theorem is valid for any similar plane figures (see Fig. 7/14).

This construction is particularly useful when you wish to find ihe sue of a circle which has the equivalent area of two or more smaller circles added together

To find the diameter of e circle which has the same area as two circles. 30 mm and GO mm diameter

(Fig. 7/15) Draw a line 30 mm long. From one end erect a perpendicular. 50 mm long. The hypotenuse of the triangle thus formed is the required diameter (58.4 mm)

If you have to find the single equivalent diameter of more than two circles, reduce them in pairs until you have two. and then finally one left.

To divide a triangle ABC into thraa parts of equal area by drawing linea parallel to one of the aidas (sey BC) (Fig 7/16)

1. Bisect AC (or AB) in 0 and erect a semi-circle, centre 0. radius OA.

2. Divide AC into three equal parts AD, DE and EC and erect perpendiculars from D and E to meet the semi-circle in D. and E,.

3. With centre A, and radius AD,, draw an arc to cut AC inD,

4. With centre A, and radius AEdraw an arc to cut AC in Et.

5. From D, and E, draw lines parallel to BC to meet AB in D, and E, respectively.

Although Fig. 7/16 shows a triangle divided into three equal areas, the construction can be used for any number of equal areas.

To divide a polygon into a number of equal areaa (aay 6) by Unas drawn parallel to the sides This construction is very similar to that used for Fig. 7/16. Proceed as for the triangle and complete as shown in Fig 7/17.

Again, this construction can be used for any polygon and can be adapted to divide any polygon into any number of equal areas.

Exercises 7

(All questions originally set in Imperial units)

1. (a) Construct the triangle ABC. shown in Fig. 1A. from the information given and then construct a triangle CDA in which CD and CB are in the ratio 5 7

(b) Construct the polygon ABCDE shown in Fig. 1B. Construct by the use of radiating lines e polygon PQRST similar to ABCDE so that both polygons stand on the line PT. Southern Regions! Examinations Board

2. Fig 2 shows a sail for a model boat. Draw the figure, full size, and construct a similar shape with the side corresponding to AB 87 mm long.

3. Without the use of a protractor or set squares construct a polygon ABCDE standing on a base AB given that AB - 95 mm, BC «- 75 mm, CD — 55 mm AE - 67.5 mm, ¿.ABC - 120". L EAB - 82J», and L CDE«90°. Also construct a similar but larger polygon so that the side corresponding with AB becomos 117.5 mm Measure and state the lengths of the sides of the enlarged polygon.

Oxford L oca/ Examinations heberrechtlich geschutztes Mi

4. Make a copy of the plane figure shown in Fig. 3.

Enlarge your figure proportionally so that the base AB measures 88 mm.

University of L ondon School Examinations

DIMENSIONS IN mm

5. In the triangle ABC, AB - 82 mm. BC - 105 mm, and CA - 68 mm. Draw a triangle similar to ABC. and having an area one-fifth the area of ABC.

Oxford and Cambridge Schools Examinations Board

6. Draw a polygon ABCDEF making AB - 32 mm. BC -38 mm, CD - 50 mm, DE - 34 mm. EF - 28 mm. FA - 28 mm. AC 56 mm. AD = 68 mm. and AE -50 mm.

Construct a further polygon similar to ABCDEF but having an area larger in the ratio of 4:3. Cambridge Local Examinations

7. Construct, full size, the figure illustrated in Fig. 4 and by radial projection, superimpose about the same centre a similar figure whoso area is three times es great as that shown in Fig. 4.

8. Fig. 5 shows a section through a length of moulding. Draw an enlarged section so that the 118 mm dimensions becomes 172 mm. Oxford Local Examinations (See Ch. 11 for informs-

9. Fig. 6 shows a shaped plate, of which DE is a quarter of an ellipse. Drew:

(a) the given view, full size;

(b) an enlarged view of the plate so that AB becomes 150 mm and AC 100 mm The distances parallel to AB and AC are to be enlarged proportionately to the increase in length of AB and AC respectively. Oxford Local Examinations (See Ch. 11 for informa-tion not in Ch. 7)

10. The shape is shown in Fig. 7 of a plate of uniform thickness

Draw the figure and. with one comer in the position shown, add a square which would represent the position of a square hole reducing the weight of the plate by 25%

11. Construct a regular hexagon having a distance between opposite sides of 100 mm. Reduce this hexagon to a square of equal area Measure and state the length of side of this square

12. A water mam is supplied by two pipes of 75 mm and 100 mm diameter. It is required to replace the two pipes with one pipe which is large enough to carry the same volume of water

Part 1 Draw the two pipes and then, using a geometrical construction, draw the third pipe. Part 2. Draw a pipe equal in area to the sum of the three pipes.

13. Three squares have side lengths of 25 mm, 37.5 mm and 50 mm respectively Construct, without resorting to calculations, a single square equal in area to the three squares, and measure and state the length of its side.

Cambridge Locel Examinations

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