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879 A square and a rectangle are each enclosed by a chain 100 ft. long, and the rectangle is 20 ft. wide. Compare their areas.

880 Find the dimensions of a rectangle whose area is 540 sq. in., and whose base and altitude are as 3: 5.

881 Find the area of an isosceles trapezoid in which the bases are 66 ft. and 84 ft., and a leg is 41 ft.

882 Find the area of a trapezoid in which the bases are 20 ft. and 34 ft., the legs 15 ft. and 13 ft.

883 Find the area of a trapezium ABCD in which AB = 7, AD = 15, BC = 13, DC = 11, and BD = 20.

884 Find the side of a square equivalent to a trapezium whose sides in order are 6, 25, 8, 35, and whose shortest diagonal is 29.

885 Two similar polygons together contain 195 sq. ft., and the ratio of similitude is 4 : 7. Find the area of each.

886 The ratio of two similar polygons is 1 : 4. If the shortest side of the first is 5, find the shortest side of the second.

887 The base of a triangle is 12 ft. Find the length of the line parallel to the base, and which bisects the triangle.

888 The side of a square is a.

large.

Find the side of a square m times as

889 The side of a square is 10 ft. Find the side of a square 5 times as large.

890 Can the altitude of a rhombus be equal to the base?

Compute S, R, and r in the following triangles in which the sides are : 891 12, 55, 65; 15, 28, 41; 19, 20, 37.

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THEOREMS

901 The square on the diagonal of a square is double the given square. 902 The square on a line is four times the square on half the line.

903 The square of either leg of a right triangle is equal to the product of the sum and difference of the other two sides.

904 If one acute angle of a right triangle is double the other, the square of one leg is three times the square of the other.

905 The square on the hypotenuse of a right triangle is equivalent to four times the square on the median to the hypotenuse.

906 The square of the altitude of an equilateral triangle is equal to three times the square of half the side.

907 Three times the square of the side of an equilateral triangle is equal to four times the square of the altitude.

908 If two triangles have the same base and their vertices in a line parallel to the base, they are equivalent.

909 A median divides a triangle into two equivalent triangles.

910 The diagonals of a parallelogram divide it into four equivalent triangles.

911 A parallelogram is bisected by any straight line drawn through the mid-point of a diagonal.

912 In the parallelogram ABCD, E, the mid-point of BC, is joined to A and D. Prove the triangles AEB and DEC equivalent.

913 If squares are constructed on two adjacent sides of a rectangle, the rectangle of their diagonals is twice the given rectangle.

914 If squares are constructed on two adjacent sides of a rectangle, the sum of the squares on their diagonals is equivalent to twice the square on the diagonal of the rectangle.

915 A trapezoid is divided into four triangles by its diagonals, two of which are similar, the other two are equivalent.

916 If the mid-points of the adjacent sides of a quadrilateral are joined, the parallelogram thus formed is equivalent to half the quadrilateral.

917 If through the vertices of a quadrilateral parallels are drawn to the diagonals, the parallelogram thus formed is equivalent to twice the quadrilateral.

918 If a line bisects the bases of a trapezoid, it bisects the trapezoid.

919 If similar polygons are constructed on the sides of a right triangle as homologous sides, the polygon on the hypotenuse is equivalent to the sum of the polygons on the legs.

920 If P is any point in the circumference of a circle whose diameter is AB, the sum of the squares on PA and PB is constant.

921 If two equal circles so intersect that the circumference of each passes through the center of the other, the square on their common chord is equivalent to three times the square on the radius.

922 The sum of the squares of the sides of a parallelogram is equal to the sum of the squares of the diagonals.

923 If the mid-points of two adjacent sides of a parallelogram are joined, the triangle thus formed is one eighth of the

parallelogram.

924 If AB, BC, and CA of the triangle ABC are each produced its own length to B', C', and A' respectively, the triangle A'B'C' is seven times the triangle ABC.

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925 If the sides of a parallelogram ABCD are produced in order, each its own length, to A', B', C', D', A'B'C'D' is a parallelogram five times the parallelogram ABCD.

926 The square inscribed in a sector which is one fourth of a circle is five eighths of the square inscribed in the semicircle.

[Let be a side of the square in the sector and s a side of the square in the semicircle whose radius is

(2).

9

$2 5 s2
4 4
.. 8 q2 = 5 s2 ; or q2 = = s.]
s2; =

r. Then 2 2 q2 (1).

Also r2

Also r2 s2 +

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927 The square inscribed in a semicircle is two fifths of the square inscribed in the circle.

928 If two equivalent triangles have an angle in each equal, the including sides are inversely proportional.

929 If from the mid-point of the base of a triangle lines are drawn parallel to the other two sides, the parallelogram thus formed is equivalent to half the triangle.

NOTE. Exercises 930-939 relate to the annexed figure, repeated from § 418.

930 The points E, B, F, are in a straight line.

931 AD and CG are parallel.

932 CE and BK are perpendicular lines.

933 BC produced bisects FH at Q, and CQ equals the half of AB.

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934 The sum of the perpendiculars dropped from E and F to AC produced is equal to AC.

935 The sum of the angles HCF and KAE is equal to three right angles.

936 If DG, EK, and FH are joined, each of the triangles DBG, EAK, and FCH is equivalent to the triangle ABC.

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940 If two equivalent triangles are on opposite sides of the same base, the common base bisects the line joining their vertices.

941 If the medians of a triangle ABC intersect in O, the triangle BOC is one-third the triangle ABC.

942 Four times the sum of the squares of the medians of any triangle is equivalent to three times the sum of the squares of the sides.

943 In a triangle whose base is b and altitude a, prove that the side of ab the inscribed square is equal to a+b

944 In the annexed figure, ABCD is a square; E, F, G, and I are the mid-points of the sides. Prove IJKL a square equivalent to one fifth of the square ABCD.

A

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I

E

945 The sum of the squares of the four sides of a quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining B the mid-points of the diagonals.

F

946 In a quadrilateral the lines which join the mid-points of the opposite sides and the line which joins the mid-points of the diagonals meet in a point of bisection.

PROPOSITION XI. PROBLEM

423 To construct a square equivalent to the sum of two given squares.

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DATA. H and K are two given squares.

REQUIRED. To construct a square equivalent to H+K.

R

SOLUTION

Construct the rt. A ABC whose legs are respectively equal to the sides of the squares H and K.

Construct the square R whose side is equal to the hypotenuse AC.

CONCLUSION. R is the square required.

PROOF.

AC2

AB2 + BC2.

Q. E. F.

"The square on the hypotenuse of a rt. ▲ is equivalent to the sum of the squares on the other two sides."

.. R≈ H+K.

§ 418

Q. E. D.

EXERCISES

947 Construct a square equivalent to the sum of two squares whose sides are 3 and 4.

948 Construct a square equivalent to the sum of two squares whose sides are 8 and 15.

949 Compute the side of a square whose area is equal to the areas of two squares whose sides are 13 in. and 84 in.

950 Compute the side of a square whose area is equal to the areas of two squares whose sides are 20 in. and 99 in.

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