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382. REMARK. - To divide a polygon into n equivalent parts by lines passing through a point P, one of the following two methods is usually used :
(a) Transform the figure into a triangle having one vertex at P. Divide the triangle into n equal parts, and transform the parts thus obtained so as to form parts of the original figure.
or, (b) Divide the figure into n parts by any method, and transform the parts thus obtained so as to fulfil the given conditions.
The two methods are illustrated by the following exercises :
Ex. 852. To divide quadrilateral ABCD into three equal parts by straight lines passing through A.
(a) Transform ABCD into A ADE.
Divide A ADE into the three equivalent parts ADF, AFG, and AGE.
As the last two parts do not lie en- D tirely in the given quadrilateral, draw GHII CA.
Then AFG - AFCH, and AF and AH are the required lines,
or, (b) Trisect DB. Draw AF, AE, CF, and CE.
Then the broken lines AFC and AEC divide the figure into three equivalent parts. To transform these parts so as to fulfil the conditions, draw FH and EK parallel to AC.
D AH and AK are the required lines.
Ex. 853. To bisect a trapezoid by a line drawn from a point P in the smaller base.
Ex. 854. To bisect a triangle by a line drawn from a point P in the base.
Ex. 855. To bisect a triangle by a line drawn perpendicular to the base.
Ex. 856. To bisect a triangle by a line parallel to a given line.
Ex. 857. A straight line passing through the midpoint of a diagonal of a parallelogram divides the figure into two equivalent parts.
Ex. 858. The area of a circumscribed polygon is equivalent to one-half the product of perimeter and radius.
Ex. 859. If through any point in a diagonal of a parallelogram, parallels are drawn to the sides, four parallelograms are formed, of which the two which do not contain the diagonal are equivalent.
Ex. 860. If any point within a parallelogram be joined to the four vertices, the sum of either pair of opposite triangles is equivalent to one-half the parallelogram.
Ex. 861. The lines joining the midpoints of the sides of a quadrilateral in succession form a parallelogram equivalent to one-half the quadrilateral.
Ex. 862. The areas of two similar triangles are to each other as the squares of any two homologous bisectors.
Ex. 863. The areas of two similar triangles are to each other as the squares of any two homologous medians.
Ex. 864. If E is any point in the diagonal AC of the parallelogram ABCD, prove that A AEBA ADE.
* Ex. 865. A quadrilateral is equivalent to a triangle if its diagonals and the angle included between them are respectively equal to two sides and the included angle of the triangle.
* Ex. 866. Two quadrilaterals are equivalent if the diagonals and the included angle of one are equal, respectively, to the diagonals and the included angle of the other. (Ex. 865.)
* Ex. 867. If two chords intersect within a circle at right angles, the sum of the squares upon their segments is constant.
Ex. 868. In any quadrilateral the sum of the squares of the four sides is equal to the sum of the squares of the diagonals, increased by four times the square of the line joining the midpoints of the diagonals.
Ex. 869. If, on two sides of triangle ABC, parallelograms DB and BG are constructed, their sides
H DE and GH be produced to meet in F, and on AC a parallelogram be constructed, having AK equal and parallel to FB, then the paral
D lelogram AL is equivalent to parallelogram AE plus parallelogram BG. (Pappus's Theorem.)
Ex. 870. Find a similar proposition for triangles constructed on
K the three sides of a given triangle.
Ex. 871. Prove the Pythagorean Theorem by means of Ex. 869.
PROBLEMS OF COMPUTATION
Ex. 872. The side of an equilateral triangle is 10 in. Find the area.
Ex. 873. Find the area of an isosceles triangle if the base is 6 and the arm 5.
Ex. 874. Find the area of a trapezoid whose bases are 9 and 11 respectively, and whose altitude is 12 ft.
Ex. 875. Find the area of a rhombus whose diagonals are 9 and 10 ft. respectively.
Ex. 876. Find the area of quadrilateral ABCD if AB = 39, BC = 52, CD= 25, AD = 60, diagonal AC = 65.
Ex. 877. A side of equilateral triangle ABC is 8. Find the side of an equilateral triangle equivalent to three times triangle ABC.
Ex. 878. The perimeter of a rectangle is 20 m., one side is 6 m. Find the area.
Ex. 879. What is the side of a square whose area is 900 sq. m. ?
Ex. 880. The area of a rhombus is equal to m, and one diagonal is equal to d. Find the other diagonal.
Ex. 881. The area of a trapezoid is 400 sq. m., its altitude is 8 m, Find the length of the line joining the midpoints of the non-parallel sides.
Ex. 882. The hypotenuse of a right triangle is 20, and the projection of one arm upon the hypotenuse is 4. What is its area ?
Ex. 883. The base and altitude of a triangle are 12 and 20 respectively. At a distance of 6 from the base, a parallel is drawn to the base. Find the areas of the two parts of the triangle.
Ex. 884. Find the area of a rectangle having one side equal to 6 and a diagonal equal to 10.
Ex. 885. Find the area of a polygon whose perimeter equals 20 ft., circumscribed about a circle whose radius is 3 ft.
Ex. 886. Find the side of an equilateral triangle equivalent to a parallelogram, whose base and altitude are 10 and 15 respectively.
Ex. 887. The sides of two equilateral triangles are 13 and 12 respectively. Find the side of an equilateral triangle equivalent to their difference.
Ex. 888. Two similar polygons have two homologous sides equal to 7 and 24 respectively. Find the homologous side of a third polygon, similar to the given polygons and equivalent to their sum.
Ex. 889. The sides of a triangle are as 8:15:17. Find the sides if the area is 480 sq. ft.
Ex. 890. The sides of a triangle are 8, 15, and 17. Find the radius of the inscribed circle.
Ex. 891. The sides of a triangle are 6, 7, and 8 ft. Find the areas of the two parts into which the triangle is divided by the bisector of the angle included by 6 and 7.
Ex. 892. Find the area of an equilateral triangle whose altitude is equal to h.
PROBLEMS OF CONSTRUCTION
Ex. 893. To construct a triangle equivalent to the sum of two given triangles.
Ex. 894. To transform a rectangle into another one, having given one side.
Ex. 895. To construct a triangle equivalent to the difference of two given parallelograms.
Ex. 896. To transform a square into an isosceles triangle, having a given base.
Ex. 897. To transform a rectangle into a parallelogram, having a given diagonal.
Ex. 898. To divide a triangle into three equivalent parts by lines drawn through a point in one of the sides.
Ex. 899. To bisect a parallelogram by a line perpendicular to a side.
Ex. 900. To bisect a parallelogram by a line perpendicular to a given line.
Ex. 901. To divide a parallelogram into three equivalent parts by lines drawn through a vertex.
Ex. 902. To bisect a trapezoid by a line drawn through a vertex.
* Ex. 903. Divide a triangle into three equivalent parts by lines drawn from a point P within the triangle.
Ex. 904. Divide a pentagon into four equal parts by lines drawn through one of its vertices.
Ex. 905. Divide a quadrilateral into four equal parts by lines drawn from a point in one of its sides.
Ex. 906. Find a point within a triangle such that the lines joining the point to the vertices shall divide the triangle into three equivalent parts.
Ex. 907. Construct a square that shall be to a given triangle as 5 is to 4.
Ex. 908. Construct a square that shall be to a given triangle as m is to n, when m and n are two given lines.
Ex. 909. Construct an equilateral triangle, that shall be to a given rectangle as 4 is to 5.
Ex. 910. Find a point within a triangle such that the lines joining the point with the vertices shall form three triangles, having the ratio 3:4:5.
Ex. 911. Divide a given line into two segments such that one segment is to the line as V2 is to v5.
Ex. 912. To transform a triangle into a right isosceles triangle.
Ex. 913. Construct a triangle similar to a given triangle and equivalent to another given triangle.
*Ex. 914. Bisect a trapezoid by a line parallel to the bases.