PROBLEMS OF CONSTRUCTION

435. To divide a triangle into three equivalent triangles by straight lines from one of the vertices to the side oppo site.

436. To construct an isosceles triangle equivalent to any given triangle, and having the same base.

437. On a given side, to construct a triangle equivalent to any given triangle.

438. Having given an angle and one of the including sides, to construct a triangle equivalent to a given triangle. 439. To construct a right triangle equivalent to a given triangle.

440. To construct a right triangle equivalent to a given triangle, and having its base equal to a given line.

441. On a given hypotenuse to construct a right triangle equivalent to a given triangle. When is the problem impossible?

442. To draw a straight line through the vertex of a given triangle so as to divide it into two parts having the ratio 2 to 5.

443. To bisect a triangle by a straight line drawn from a given point in one of its sides.

$398 444. On a given side to construct a rectangle equivalent to a given square.

445. To construct a square equivalent. to a given triangle.

446. To construct a square equivalent to the sum of two given triangles.

447. On a given side to construct a rectangle equivalent to the sum of two given squares.

448. To construct a square which shall have a given ratio to a given hexagon.

449. Through a given point within any parallelogram to draw a straight line dividing it into two equivalent parts.

PROBLEMS FOR COMPUTATION

450. (1.) Find the area of a parallelogram one of whose sides is 37.53 m., if the perpendicular distance between it and the opposite side is 2.95 dkm.

(2.) Required the area of a rhombus if its diagonals are in the ratio of 4 to 7, and their sum is 16.

(3.) In a right triangle the perpendicular from the vertex of the right angle to the hypotenuse divides the hypotenuse into the segments m and n. Find the area of the triangle. (4.) If the hypotenuse of an isosceles right triangle is 30 ft., find the number of ares in its area.

(5.) Find the area of an isosceles right triangle if the hypotenuse is equal to a.

(6.) If one of the equal sides of an isosceles triangle is 17 dkm. in length and its base is 30 m., find the area of the triangle.

(7.) Find the area of an isosceles triangle if one of the equal sides is a and its base is b.

(8.) If in the above example a= 17.163 hm. and b=27.395 hm., how many acres are there in the triangle?

(9.) Find the area of an equilateral triangle if one of the sides equals 16 m.

(10.) If the side of an equilateral triangle is a, find its

area.

(11.) If each side of a triangular park measures 196.37 rds., how many hectares does it contain?

(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.

P

h

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

h =2 √√√s(s—a)(s—b)(s—c).

But

C

Whence

area=

x2 √√s(s—a)(s—b)(s—c)

$ 373(31)

= s(s− a)(s—b)(s—c).

(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 ra dius of the inscribed circle.

a

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