PROBLEM 16.

To draw squares whose areas shall be respectively twice, three-times, four-times, that of a given square.

Hence find graphically approximate values of √2, √3, √4, √5, ....

From OX mark off OB equal to PA, and join PB; then PB2 OP2 + OB2 = 1 + 2 = 3.

=

.. PB = √√√3.

From OX mark off OC equal to PB, and join PC; then PC2 = OP2 + OC2 = 1+3=4.

... PC=√√√4.

The lengths of PA, PB, PC may now be found by measurement; and by continuing the process we may find √5, √6, √7, ....

EXERCISES ON THEOREMS 29, 30 (Continued).

13. Prove the following formula :

Diagonal of square = side × √2.

Hence find to the nearest centimetre the diagonal of a square on a side of 50 metres.

Draw a plan (scale 1 cm. to 10 metres) and obtain the result as nearly as you can by measurement.

14. ABC is an equilateral triangle of which each side=2m units, and the perpendicular from any vertex to the opposite side=p.

Prove that p=m√3.

Test this result graphically, when each side=8 cm.

15. If in a triangle a=m2 – n2, b=2mn, c=m2+n2; prove algebraically that c2=a2+b2.`

Hence by giving various numerical values to m and n, find sets of numbers representing the sides of right-angled triangles.

16. In a triangle ABC, AD is drawn perpendicular to BC. Let p denote the length of AD.

(i) If a=25 cm., p=12 cm., BD=9 cm.; find b and c.

(ii) If b=41", c=50′′, BD=30"; find p and a.

And prove that √b2 − p2 +√c2 − p2=a.

17. In the triangle ABC, AD is drawn perpendicular to BC. Prove that c2 - BD2=62 – CD2.

If a=51 cm., b=20 cm., c=37 cm.; find BD.

Thence find p, the length of AD, and the area of the triangle ABC.

18. Find by the method of the last example the areas of the triangles whose sides are as follows:

(i) a=17", b=10", c=9′′. (ii) a=25 ft., b=17 ft., c=12 ft. (iii) a=41 cm., b=28 cm., c=15 cm. (iv) a = 40 yd., b=37 yd., c=13 yd.

19. A straight rod PQ slides between two straight rulers OX, OY placed at right angles to one another. In one position of the rod OP=5.6 cm., and ŎQ=3·3 cm. If in another position OP=40 cm, find OQ graphically; and test the accuracy of your drawing by calculation.

20. ABC is a triangle right-angled at C, and p is the length of the perpendicular from C on AB. By expressing the area of the triangle in two ways, shew that

Hence deduce

pc=ab.

PROBLEMS ON AREAS.

PROBLEM 17.

To describe a parallelogram equal to a given triangle, and having one of its angles equal to a given angle.

Let ABC be the given triangle, and D the given angle. It is required to describe a parallelogram equal to ABC, and having one of its angles equal to D.

Construction.

Proof.

Bisect BC at E.

At E in CE, make the CEF equal to D;

through A draw AFG par1 to BC; and through C draw CG par1 to EF. Then FECG is the required parm.

Join AE.

Now the ABE, AEC are on equal bases BE, EC, and of the same altitude;

.. the ABC is double of the AEC.

But FECG is a parm by construction ;
and it is double of the AEC,

being on the same base EC, and between the same par1s EC and AG.

EXERCISES.

(Graphical.)

1. Draw a square on a side of 5 cm., and make a parallelogram of equal area on the same base, and having an angle of 45°.

Find (i) by calculation, (ii) by measurement the length of an oblique side of the parallelogram.

2.

Draw any parallelogram ABCD in which AB=21′′ and AD=2"; and on the base AB draw a rhombus of equal area.

DEFINITION. In a parallelogram ABCD, if through any point K in the diagonal AC parallels EF, HG are drawn to the sides, then the figures EH, GF are called parallelograms about AC, and the figures EG, HF are said to be their complements.

A H

E

B

G

3. In the diagram of the preceding definition shew by Theorem 21 that the complements EG, HF are equal in area.

Hence, given a parallelogram EG, and a straight line HK, deduce a construction for drawing on HK as one side a parallelogram equal and equiangular to the parallelogram EG.

4. Construct a rectangle equal in area to a given rectangle CDEF, and having one side equal to a given line AB.

If AB=6 cm., CD=8 cm., CF=3 cm., find by measurement the remaining side of the constructed rectangle.

5. Given a parallelogram ABCD, in which AB=2·4′′, AD=1·8′′, and the LA=55°. Construct an equiangular parallelogram of equal area, the greater side measuring 2.7". Measure the shorter side.

Repeat the process giving to A any other value; and compare your results. What conclusion do you draw?

6. Draw a rectangle on a side of 5 cm. equal in area to an equilateral triangle on a side of 6 cm.

Measure the remaining side of the rectangle, and calculate its approximate area.

PROBLEM 18.

To draw a triangle equal in area to a given quadrilateral.

Let ABCD be the given quadrilateral.

It is required to describe a triangle equal to ABCD in area.

Through C draw CX par1 to DB, meeting AB produced in X. Join DX.

Proof.

Then DAX is the required triangle.

Now the XDB, CDB are on the same base DB and

between the same par1s DB, CX;

.. the XDB = the CDB in area.

To each of these equals add the ADB ;

then the DAX = the fig. ABCD.

COROLLARY. In the same way it is always possible to draw a rectilineal figure equal to a given rectilineal figure, and having fewer sides by one than the given figure; and thus step by step, any rectilineal figure may be reduced to a triangle of equal area.

For example, in the adjoining diagram the five-sided fig. EDCBA is equal in area to the four-sided fig. EDXA.

The fig. EDXA may now be reduced to an equal

DXY.

E

A

B X