(12.) If the perimeter of an equilateral triangle is 523.65

ft., find its area.

(13.) Find the area of a triangle, if two of its sides are 6 in. and 7 in. and the included angle is 30°.

(14.) Show that, if a and b are the sides of a triangle, the area is ab, when the included angle is 30° or 150°; ‡ ab√2, when the included angle is 45° or 135°; ab√3, when the included angle is 60° or 120°.

(15.) Find the area of a triangle, if two of its sides are 43.746 mm. and 15.691 mm., and the included angle is 120°. (16.) How many square feet are there in the entire surface of a house 50 ft. long, 40 ft. wide, 30 ft. high at the corners, and 40 ft. high at the ridge-pole?

(17.) Find the area of a triangle whose sides are a, b, and c.

Solution.—The area of the triangle ABC==×h.

(18.) Find the area of a triangle whose sides are 119.3 m., 147.35 m., and 7 dkm.

(19.) Required the area of the quadrilateral ABCD, if the four sides AB, BC, CD, and DA measure respectively 63.57, 113.29, 39.637, and 156 ft., and the diagonal AC=150.26 ft.

(20.) If the bases of a trapezoid are respectively 97 m. and 133 m., and its area is 46 ares, find its altitude.

(21.) Find the area of a trapezoid of which the bases are 73 ft. and 57 ft., and each of the other sides is 17 ft.

(22.) Find the area of a trapezoid of which the bases are a and b and the other sides are each equal to d.

(23.) If in the triangle ABC a line MN is drawn parallel to the side AC so that the smaller triangle which it cuts off equals one-third of the whole triangle, find MN in terms of AC.

(24) Through a triangular field a path runs from one corner to a point in the opposite side 204 yds. from one end, and 357 yds. from the other. What is the ratio of the two parts into which the field is divided?

(25.) If a square and a rhombus have equal perimeters, and the altitude of the rhombus is four-fifths its side, compare the areas of the two figures.

(26.) The altitude upon the hypotenuse of an isosceles right triangle is 3.1572 m. Find the side of an equivalent

square.

(27.) If the areas of two triangles of equal altitude are 9 hectares and 324 ares respectively, what is the ratio of their bases?

(28.) A triangle and a rectangle are equivalent. (a.) If their bases are equal find the ratio of their altitudes. (b.) Compare their bases if their altitudes are equal.

(29.) Two homologous sides of two similar polygons are respectively 12 m. and 36 m. in length, and the area of the first is 180 sq. m. What is the area of the second?

(30.) Two similar fields together contain 579 hectares. What is the area of each if their homologous sides are in the ratio of 7 to 12?

(31.) In a triangle having its base equal to 24 in. and an area of 216 sq. in., a line is drawn parallel to the base through a point 6 in. from the opposite vertex. Find the area of the smaller triangle thus formed.

(32.) The altitude of a triangle is a and its base is b; the altitude, homologous to a, of another triangle, similar to the first, is c. Find the altitude, base, and area of a triangle similar to the given triangles and equivalent to their sum. (33.) Construct a square equivalent to the sum of the squares whose sides are 20, 16, 9, and 5 cm.

(34.) If the sides of a triangle are 113.61 cm., 97.329 cm., and 82.52 cm., find the areas of the parts into which it is divided by the bisector of the angle opposite the first side.

(35.) If to the base b of a triangle the line d is added, how much must be taken from its altitude h that its area may remain unchanged?

(36.) If the sides of a triangle are a, b, and c, find the radius of the inscribed circle.

B

Solution.—The area of the triangle CBP=2xr. b

The area of the triangle CAP=-xr.

The area of the triangle BAP=2xr.

The sum of these areas, or the area of the triangle ABC,

(37.) If the sides of a triangle are 173.52 cm., 125.3 cm., and 96.357 cm., find the radius of the inscribed circle.

PLANE GEOMETRY

BOOK V

REGULAR POLYGONS AND CIRCLES.

SYMMETRY WITH RE

SPECT TO A POINT

451. Defs.-A figure turns half-way round a point, if a straight line of the figure passing through the point turns through 180°, i. e., half of 360°.

A figure turns one-third-way round a point, if a straight line of the figure passing through the point turns through 120°, i. e., one-third of 360°.

In general, a figure turns one-nth way round a point if a straight line of the figure passing through the point turns through one-th of 360°.

452. Exercise.-If a figure is turned half-way round on a point as a pivot, i. e., so that one straight line of the figure passing through that point turns through 180°, prove that every other straight line of the figure passing through that point turns through 180°.

453. Exercise.-In the same case, prove that every straight line not passing through the pivot makes after the rotation. an angle of 180° with its original position.

454. Exercise.-If a figure turns one-third way round, prove that every straight line, whether passing through the pivot or not, makes after the rotation an angle of 120° with its original position.