Plane and Solid Geometry

PROPOSITION XXI. Problem.

287. To construct a polygon similar to a given polygon and equal in area to another given polygon.

Let A and B be two given polygons.

To construct a polygon similar to A, and equal in area to B. SUG. 1. Construct squares equal in area to the polygons A and B respectively, and let m and n represent the sides of the squares.

SUG. 2. Let a represent one side of the polygon A, and find a fourth proportional to m, n and a.

SUG. 3. Let p represent the fourth proportional just found, and construct a polygon P similar to A, having the side in the polygon P homologous to the side a in the polygon A.

SUG. 4. Compare the ratio

with the ratio

Compare

with

(See exercise 129.) Compare the ratios

288. A line is divided in mean and extreme ratio when the ratio of the whole line to the greater segment equals the ratio of the greater to the lesser segment.

PROPOSITION XXII. PROBLEM.

289. To divide a straight line in mean and extreme

Let A B be a given straight line.

To divide AB in mean and extreme ratio.

SUG. 1. At B, erect a 1 to A B, and on this lay off BOA B. AB.

With O as a center, and O B as a radius, describe a circumference. Draw a line through A and O, meeting the circumference at N and M.

On A B lay off a distance A X equal to Á N.

PROPOSITIONS IN CHAPTER IV.

PROPOSITION I.

Two parallelograms having equal bases and equal altitudes are equal in area.

PROPOSITION II.

If two rectangles have equal altitudes, the ratio of their areas equals the ratio of their bases.

PROPOSITION III.

The number of units of area in any rectangle is equal to the prod uct of the number of linear units in the base and altitude.

PROPOSITION IV

The area of a parallelogram equals the product of its base and altitude.

PROPOSITION V.

The area of a triangle equals one half the product of its base and altitude.

PROPOSITION VI.

The area of a trapezoid equals one half the product of its altitude by the sum of its bases.

PROPOSITION VII.

The ratio of the areas of two similar triangles is equal to the ratio of the squares of their homologous sides, or homologous altitudes.

PROPOSITION VIII.

The ratio of the areas of two similar polygons equals the ratio of the squares of two homologous sides.

PROPOSITION IX.

The square described upon the hypotenuse of a right triangle is equal to the sum of the squares described upon the other two sides.

PROPOSITION X.

The square described on the sum of two lines equals the sum of the squares described on the two lines plus twice the rectangle whose sides are the two lines.

PROPOSITION XI.

The squares described on the difference of two lines equals the sum of the squares described on the two lines minus twice the rectangle whose sides are the two lines.

PROPOSITION XII.

The square on the side opposite an acute angle of a triangle equals the sum of the squares on the other two sides minus twice the product of one side by the projection of the other side upon that side.

PROPOSITION XIII.

The square on the side opposite an obtuse angle of a triangle equals the sum of the squares on the other two sides plus twice the product of one of the sides by the projection of the other side upon that side.

PROPOSITION XIV.

Upon a given line as base, to construct a rectangle equal in area to a given square.

PROPOSITION XV.

To construct a square equal in area to a given rectangle.

PROPOSITION XVI.

To construct a square whose area is equal to the sum of the areas of two given squares.

PROPOSITION XVII.

To construct a square whose area is equal to the difference of the areas of two given squares.

PROPOSITION XVIII.

To construct a rectangle in which the sum of the base and altitude equals a given line and the area equals the area of a given square