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Substituting in (3), and extracting the square root,

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the plus sign, only, being used, since A < 90°; hence, as A represents any angle,

The cosine of half of any angle of a plane triangle, is equal to the square root of the product of half the sum of the three sides, and half that sum minus the side opposite the angle, divided by the rectangle of the adjacent sides.

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If we subtract each member of equation (2) from 1, and recollect that 1 cos A = 2 sin2 A (Art. 66), we have

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Substituting in (5) and reducing, we have

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The sine of half an angle of a plane triangle, is equal to the square root of the product of half the sum of the three sides minus one of the adjacent sides and half that sum minus the other adjacent side, divided by the rectangle of the adjacent sides.

Applying logarithms, we have

log sin A = [log (†s —

--

b) + log (sc) + (a. c.) log b
+ (a. c.) log c]. (B.)

Third Case. To find the area of a triangle when the three sides are given.

Let ABC represent a triangle whose sides a, b, and c are given. From the principle demonstrated in the last case, we have

Q = bc sin A.

But, from formula (A'), Trig., Art. 66, we have

B

a

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whence,

Q = bc sin A cos A.

Substituting for sin A and cos A, their values, taken from Lemma, and reducing, we have

Q = √ts (įs — a) (is — b) (4s — c);

hence, we may write the following

RULE.-Find half the sum of the three sides, and from it subtract each side separately. Find the continued product of the half sum and the three remainders, and extract its square root; the result will be the area required.

It

is generally more convenient to employ logarithms; for this purpose, applying logarithms to the last equation, we have

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[log is + log (s—a) + log (†s—b) + log (†s—c)];

hence, we have the following

RULE.-Find the half sum and the three remainders as before, then find the half sum of their logarithms; the number corresponding to the resulting logarithm will be the area required.

Examples.

1. Find the area of a triangle, whose sides are 20, 30, and 40.

=

We have s 45, js — a = 25, fs — b = 15, sc = 5. By the first rule,

Q:
Q = √45 x 25 x 15 x5 = 290.4737, Ans.

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2. How many square yards are there in a triangle, whose sides are 30, 40, and 50 feet?

Ans. 663.

To find the area of a trapezoid.

98. From the principle demonstrated in Book IV., Prop. VII., we may write the following

RULE.-Find half the sum of the parallel sides, and multiply it by the altitude; the product will be the area required.

Examples.

1. In a trapezoid the parallel sides are 750 and 1225, and the perpendicular distance between them is 1540; what is the area? Ans. 1520750.

2. How many square feet are contained in a plank, whose length is 12 feet 6 inches, the breadth at the greater end 15 inches, and at the less end 11 inches? Ans. 13.

3. How many square yards are there in a trapezoid, whose parallel sides are 240 feet, 320 feet, and altitud 66 feet?

Ans. 20531 S

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To find the area of any quadrilateral.

99. From what precedes, we deduce the following

RULE. Join the vertices of two opposite angles by a diagonal; from each of the other vertices let fall perpendiculars upon this diagonal; multiply the diagonal by half of the sum of the perpendiculars, and the product will be the area required.

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2. How many square yards of paving are there in the quadrilateral, whose diagonal is 65 feet, and the two perpendiculars let fall on it 28 and 33 feet?

To find the area of any polygon.

Ans. 2221.

100. From what precedes, we have the following

RULE.-Draw diagonals dividing the proposed polygon into trapezoids and triangles: then find the area of these figures separately, and add them together for the area of the whole polygon.

Example.

1. Let it be required to determine the area of the polygon ABCDE, having five sides.

Let us suppose that we have measured the diagonals and perpendiculars,

E

d

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