CONSTRUCTIONS

PROBLEM 1

To construct a parallelogram equal in area to a given triangle and having one of its angles equal

Let ABC be the given ▲ and D the given 2.

It is required to construct a gm equal in area to ▲ ABC and having one ▲ equal to ▲ D.

Construction.-Through A draw AFG|| BC.

Bisect BC

at E. At E make CEF = L D. Through c draw

CG || EF.

FC is the required gm.

Proof-Draw any line HK

to the two st. lines.

HK is the common altitude of the gm FC and the

PROBLEM 2

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

D

B

Let ABCD be the given quadrilateral.

It is required to construct a ▲ equal in area to

ABCD.

Construction. Join AC. Through D draw DE || AC and meeting BC produced at E. Join AE.

PROBLEM 3

To construct a triangle equal in area to a given rectilineal figure.

A

Let the pentagon ABCDE be the given rectilineal figure.

Construction.- Join AD, BD. Through E, draw EF AD and meeting BA at F. Through C draw CG || BD and meeting AB at G.

By this method a ▲ may be constructed equal in area to a given rectilineal figure of any number of sides; e.g., for a figure of seven sides, an equivalent figure of five sides may be constructed, and then, as in the construction just given, a ▲ may be constructed equal to the figure of five sides.

PROBLEM 4

To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given angle.

Let ABCDE be the given rectilineal figure and F the given 4.

It is required to construct a gm having an = L F.

= ABCDE, and

Construction.-Make A DMH equal in area to figure

ABCDE.

Make gm LGHK = DMH, and having LGH =

L F.

(II-Prob. 3, p. 112.)

(II-Prob. 1, p. 110.)

LGH =

Then gm LGHK = figure ABCDE, and has

LF.

79.-Exercises

1. Construct a rect. equal in area to a given A.

2. Construct a rect. equal in area to a given quadrilateral.

3. Construct a quadrilateral equal in area to a given hexagon.

4. On one side of a given ▲ construct a rhombus equal in area to the given ▲.

5. Construct a ▲ equal in area to a given || gm, and having one of its s = a given 2.

PROBLEM 5

To construct a triangle equal in area to a given triangle and having one of its sides equal to a given straight line.

Let ABC be the given ▲ and D the given st. line. It is required to make a ▲ = ▲ ABC and having one side = D.

Construction.-From BC, produced if necessary, cut off BE = D. Join AE. Through C draw CF | EA and meeting BA, or BA produced at F. Join FE.

.. ▲ FCE = ▲ AFC. (II-5, p. 101.)

= ▲ FBC + ▲ AFC,

:. Δ FBC + Δ FCE