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ON THE CONSTRUCTION OF TRIANGLES.

(Graphical Exercises.)

1. Draw a triangle whose sides are 7.5 cm., 62 cm., and 5.3 cm. Draw and measure the perpendiculars dropped on these sides from the opposite vertices.

[N.B. The perpendiculars, if correctly drawn, will meet at a point, as will be seen later. See page 207.]

2.

Draw a triangle ABC, having given a=3′00′′, b=2′50′′, c=2·75′′.
Bisect the angle A by a line which meets the base at X. Measure
BX and XC (to the nearest hundredth of an inch); and hence calculate
BX
the value of to two places of decimals. Compare your result with
CX

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3. Two sides of a triangular field are 315 yards and 260 yards, and the included angle is known to be 39°. Draw a plan (1 inch to 100 yards) and find by measurement the length of the remaining side of the field.

4. ABC is a triangular plot of ground, of which the base BC is 75 metres, and the angles at B and C are 47° and 68° respectively. Draw a plan (scale 1 cm. to 10 metres). Write down without measurement the size of the angle A; and by measuring the plan, obtain the approximate lengths of the other sides of the field; also the perpendicular drawn from A to BC.

5. A yacht on leaving harbour steers N. E. sailing 9 knots an hour. After 20 minutes she goes about, steering N. W. for 35 minutes and making the same average speed as before. How far is she now from the harbour, and what course (approximately) must she set for the run home? Obtain your results from a chart of the whole course, scale 2 cm. to 1 knot.

6. Draw a right-angled c=10'6 cm. and one side find the value of

2-a2.

triangle, given that the hypotenuse 56 cm. Measure the third side b; and Compare the two results.

7. Construct a triangle, having given the following parts: B=34°, b=55 cm., c=8.5 cm. Shew that there are two solutions. Measure the two values of a, and also of C, and shew that the latter are supplementary.

8. In a triangle ABC, the angle A=50°, and b=6·5 cm. Illustrate by figures the cases which arise in constructing the triangle, when (i) a=7 cm. (ii) a=6 cm. (iii) a = 5 cm. (iv) a 4 cm.

9. Two straight roads, which cross at right angles at A, are carried over a straight canal by bridges at B and C. The distance between the bridges is 461 yards, and the distance from the crossing A to the bridge B is 261 yards. Draw a plan, and by measurement of it ascertain the distance from A to C.

(Problems. State your construction, and give a theoretical proof.)

10. Draw an isosceles triangle on a base of 4 cm., and having an altitude of 6.2 cm. Prove the two sides equal, and measure them to

the nearest millimetre.

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11. Draw an isosceles triangle having its vertical angle equal to a given angle, and the perpendicular from the vertex on the base equal to a given straight line.

Hence draw an equilateral triangle in which the perpendicular from one vertex on the opposite side is 6 cm. Measure the length of a side to the nearest millimetre.

12. Construct a triangle ABC in which the perpendicular from A on BC is 5'0 cm., and the sides AB, AC are 5'8 cm. and 90 cm. respectively. Measure BC.

13. Construct a triangle ABC having the angles at B and C equal to two given angles L and M, and the perpendicular from A on BC equal to a given line P.

14. Construct a triangle ABC (without protractor) having given two angles B and C and the side b.

15. On a given base construct an isosceles triangle having its vertical angle equal to a given angle L.

16. Construct a right-angled triangle, having given the length of the hypotenuse c, and the sum of the remaining sides a and b.

If c=5·3 cm., and a+b=7·3 cm., find a and b graphically; and calculate the value of √√a2+b2.

17. Construct a triangle having given the perimeter and the angles at the base. For example, a+b+c=12 cm., B=70°, C=80°.

18. Construct a triangle ABC from the following data:

a=65 cm., b+c=10 cm., and B=60°.

Measure the lengths of b and c.

19. Construct a triangle ABC from the following data:

a=7 cm., c-b=1 cm., and B=55°.

Measure the lengths of b and c.

12.

THE CONSTRUCTION OF QUADRILATERALS.

It has been shewn that the shape and size of a triangle are completely determined when the lengths of its three sides are given. A quadrilateral, however, is not completely determined by the lengths of its four sides. From what follows it will appear that five independent data are required to construct a quadrilateral

PROBLEM 11.

To construct a quadrilateral, given the lengths of the four sides, and one angle.

[blocks in formation]

Let a, b, c, d be the given lengths of the sides, and A the angle between the sides equal to a and d.

Construction. Take any straight line AX, and cut off from it AB equal to a.

Make the BAY equal to the LA.

From AY cut off AD equal to d.

With centre D, and radius c, draw an arc of a circle.

With centre B and radius b, draw another arc to cut the former at C.

Join DC, BC.

Then ABCD is the required quadrilateral; for by construction the sides are equal to a, b, c, d, and the DAB is equal to the given angle.

PROBLEM 12.

To construct a parallelogram having given two adjacent sides and the included angle.

P

A

B

Let P and Q be the two given sides, and A the given angle.

Construction 1. (With ruler and compasses.) Take a line AB equal to P; and at ▲ make the BAD equal to the A, and make AD equal to Q.

With centre D, and radius P, draw an arc of a circle.

With centre B, and radius Q, draw another arc to cut the former at C.

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and these are alternate angles,

.. DC is par1 to AB.

Also DC AB;

Theor. 7.

.. DA and BC are also equal and parallel. Theor. 20.

Construction 2.

.. ABCD is a parTM.

(With set squares.)

Draw AB and AD as

before; then with set squares through D draw DC par1 to AB, and through B draw BC par1 to AD.

By construction, ABCD is a part having the required parts.

PROBLEM 13.

To construct a square on a given side.

A

B

Let AB be the given side.

Construction 1. (With ruler and compasses.) At A draw AX perp. to AB, and cut off from it AD equal to AB.

With B and D as centres, and with radius AB, draw two arcs curting at C.

Proof.

Join BC, DC.

Then ABCD is the required square.

As in Problem 12, ABCD may be shewn to be a parTM. And since the BAD is a right angle, the figure is a rectangle. Also, by construction all its sides are equal.

.. ABCD is a square.

Construction 2. (With set squares.) At A draw AX perp. to AB, and cut off from it AD equal to AB.

Through D draw DC par to AB, and through B draw BC par to AD meeting DC in C.

Then, by construction, ABCD is a rectangle. [Def. 3, page 56.] Also it has the two adjacent sides AB, AD equal.

it is a square.

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