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Rule. Multiply the Side by the Length of a Perpendicular let fall from one of the obtufe Angles to the Side oppofite fuch Angle.

Let A B C D reprefent à Rhombus, each of whofe Sides is 16 Feet: A Perpendicular let fall from the obtufe Angle at B, on the Side DC, will interfect it in the Point E, fo will BE be 13 Feet; and this being

D

B

C

multiplied into the given Side, the Product is the Area of the Rhombus.

16

13

48

16

208 Area,

Article 55. To find the Area of a Rhomboides.

Definition. A Rhomboides is a Figure whofe oppofite Sides and oppofite Angles are equal.

Rule. Multiply one of the longeft Sides by the Perpendicular let fall from one of the obtufe Angles on one of the longeft Sides.

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Article 56. To measure a Triangle.

Definition. A Triangle is à Figure having three Sides and three Angles. There are three Sorts of Triangles, denominated from the Angles: For, if one of the Angles of a Triangle is an exact Square, (which, by Mathematicians, is called a right Angle) fuch Triangle is called a right-angled Triangle: If

R 2

there

there is no right Angle, it is called an oblique-angled Triangle, of which there are two Kinds; if all the three Angles are acute, that is, each less than a right Angle, it is called an acute-angled Triangle; but, if one of the Angles is greater than a right Angle, it is called an obtufe-angled Triangle.

Rule. If it be a right-angled Triangle, multiply the Bafe by half the Perpendicular, or half the Bafe by the Perpendicular, and the Product is the Area: But, if it be an oblique-angled Triangle, (whether obtufe or acute) multiply half the Bafe by the Length of the Perpendicular let fall on the Bafe from the Angle oppofite to it, or half the perpendicular Height by the Bafe, and the Product is the Area. The longest Side of a Triangle is ufually called the Bafe, except in a right-angled Triangle, where the longest of the two Legs which include the right Angle is called the Bafe.

The right angled Triangle ABC, right-angled at C; the Bafe AC is 21 Feet, and the Perpendicular BC 15.5 Feet.

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B

775

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1550

162.75 Area.

The oblique-angled Triangle ABC being given, let fall a Perpendicular from the Angle at B on the Bafe A C, and that Perpendicular is the Height of the Triangle. The Bafe A C being 28.6, and the Perpendicular BD 10, to find the Area?

14.3 Half the Base.

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10 Height of the Triangle.

143.0 Area.

And here the Learner may obferve, that this Way of measuring an oblique-angled Triangle is the fame with the measuring a right-angled Triangle. In this Place, therefore, it may be proper to inftruct the Reader in one of the Pro

perties

perties of a right-angled Triangle: Which is this, that the Square of the longeft Side of a right-angled Triangle, ufually called the Hypothenufe, is equal to the Sum of the Squares of the two other Sides, ufually called the Legs. This is the forty-feventh Propofition of the firft Book of Euclid: For the finding of which Pythagoras is faid to have facrificed a Hecatomb to the Mufes, it being a Thing of fuch confiderable Ufe; for, by this Means, any two Sides of a right-angled Triangle being given, the other may be found by common Arithmetic. Thus, in the right-angled Triangle ABC, the Base AC and Perpendicular BC being given, the Hypothenuse AB may be found by extra ing the Square Root of the Sum of the Squares of the Base and Perpendicular.

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And, if the Hypothenufe and one of the Legs are given, the other may be found by fubtracting the Square of the given Leg from the Square of the Hypothenufe, for the Square Root of the Remainder is the other Leg..

In these Operatians, the true Length of the Side fought cannot always be found, because the Sum of the Squares of the Legs, or the Difference of the Squares of the Hypothenuse and given Side, rarely happen to be Square Numbers: In which Cafe, the Extraction may be continued as far as is neceffary, in order to come as near the Truth as may be judged proper. But there are fome Square Numbers whofe Sums do make a Square: Of which Sort are the Numbers 3 and 4, whose Squares being added together make 25, which is the Square of 5: Therefore, if the Bafe of a Triangle is 4, and the Perpendicular 3, the Hypothenufe will be 5: And, if any of thefe Numbers are multiplied by any other Number, thofe Products will be the Sides of right-angled Triangles; as 6, 8, 10, and 15, 20, 25. Thus Artificers, when they fet off

the

the Corner of a Building, ufually measure 6 Feet on one Side and 8 Feet on the other, then laying crofs a ten-feet Pole, it makes the Corner a true right Angle.

Article 57. There is another Method of finding the Area of Triangles, the three Sides being given.

Rule. Add the three Sides together, then take the Half of that Sum, and out of it fubtract each Side feverally: Multiply the Half of the Sum and thefe Remainders continually; then extract the Square Root of this Product, which will be the Area of the Triangle.

In the former oblique Triangle, the Bafe AC is given 28.6, the Side A B is 18.32, and the Side BC is 16.6, to find the Area?

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Thus is 142.999 the Area of the Triangle, which is within oor Part of what was before found.

Article 58. To measure a Trapezium.

Definition. A Trapezium is an irregular Figure, of four unequal Sides, and unequal Angles.

Rule, Draw a Diagonal from one of the Angles to the oppofite Angle, and then will the Trapezium be divided into two Triangles, of which the Diagonal is the common Base: Then drawing Perpendiculars from the other opposite Angles to the Diagonal, add thofe Perpendiculars together; and multiply the Half of that Sum into the Diagonal, or the Half of the Diagonal into the Sum of the Perpendiculars, and that Product is the Area of the Trapezium.

In the Trapezium ABCD, A the Diagonal AC is 498, the Perpendicular DE 10.8, and the Perpendicular BF 18.8: The Sum of the Perpendiculars is 29.6, whofe Half is 14.8, which, being multiplied into 498, gives the Area.

D

B

C

498

14.8

3984

1992

498

7370.4 Area of the Trapezium.

Artice 59. To meafure any irregular Figure.

Rule. Divide the Figure into Triangles, by drawing Diagonals from one Angle to another; then measure all the Triangles, by either of the Rules already taught at Art. 56 or 57, and add the feveral Areas of all the Triangles together, which Sum will be the Area of the given Figure.

The irregular Figure ABCDEF being given, divide it into Triangles by the Diagonals F B, EB, and DB: Then may the Triangles be measured by letting fall Perpendiculars on their respective Bafes, as Ba, Bb, Dc, Fd, and multipying those Perpendiculars by half their respective Bases, according to the Rule of Art. 56; or by measuring the Sides of each

Triangle,

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