PROPOSITION XIII. PROBLEM

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

Given squares P and Q.

To construct a square the sum of P and Q.

I. Construction

1. Construct the rt. ▲ ABC, having for its sides p and q, the sides of the given squares.

2. On r, the hypotenuse of the A, construct the square R. 3. R is the required square.

II. The proof and discussion are left to the student.

Ex. 892. Construct a square equivalent to the sum of three or more given squares.

Ex. 893. Construct a square equivalent to the difference of two squares. Ex. 894. Construct a square equivalent to the sum of a given square and a given triangle.

Ex. 895. Construct a polygon similar to two given similar polygons and equivalent to their sum. (See § 509.)

Ex. 896. Construct a polygon similar to two given similar polygons and equivalent to their difference.

Ex. 897. Construct an equilateral triangle equivalent to the sum of two given equilateral triangles.

Ex. 898. Construct an equilateral triangle equivalent to the difference of two given equilateral triangles,

PROPOSITION XIV. PROBLEM

513. To construct a polygon similar to one of two given polygons and equivalent to the other.

Given polygons P and Q, with a side of P.
To construct a polygon ~P and

I. Analysis

Q.

x

1. Imagine the problem solved and let R be the required polygon with side x homol. to s, a side of P:

2. Then P R = s2: x2; i.e. P: Q: = s2:22, since Q≈ R. (1) 3. Now to avoid comparing polygons which are not similar, we may reduce P and Q to squares. Let the sides of these squares be m and n, respectively; then m2 P and n2 Q. 4. .. m2 : n2 = s2 : x2, from (1).

5. .. m : n=s: x.

6. That is, x is the fourth proportional to m, n, and s.

II. The construction, proof, and discussion are left as an exercise for the student.

514. Historical Note. This problem was first solved by Pythagoras about 550 B.C.

Ex. 899. Construct a triangle similar to a given triangle and equivalent to a given parallelogram.

Ex. 900. Construct a square equivalent to a given pentagon.

Ex. 901.

Construct a triangle, given its angles and its area (equal to that of a given parallelogram). HINT. See Prop. XIV.

Ex. 902. Divide a triangle into two equivalent parts by a line drawn perpendicular to the base. HINT. Draw a median to the base, then apply Prop. XIV.

Ex. 903. Fig. 1 represents maps of Utah and Colorado drawn to

Ex. 906. Divide a triangle into two equivalent parts by a line drawn from a given point in one of its sides.

HINT.

Let M be the given point in AC of triangle ABC; then

AB AC AM AX, where MX is the required line.

=

Ex. 907. A represents a station. Cars approach the station on track BA and leave the station on track AC. Construct an arc of a circle DE, with given radius r, connecting the two intersecting car lines, and so that each car

line is tangent to the arc.

This same principle is involved in designing

a building between two streets forming at their A‹
point of intersection a small acute angle, as
the Flatiron Building in New York City.

Ex. 908. Find the area of a rhombus if its diagonals are in the ratio of 5 to 7 and their sum is 16.

MISCELLANEOUS EXERCISES

C

Ex. 909. Show that if a and b are two sides of a triangle, the area is ab when the included angle is 30° or 150°; 4 ab√2 when the included angle is 45° or 135°; ab√3 when the included angle is 60° or 120o.

Ex. 910. The sum of the perpendiculars from any point within a convex polygon upon the sides is constant.

HINT. Join the point with the vertices of the polygon, and consider the sum of the areas of the triangles.

Ex. 911. The sum of the squares on the segments of two perpendicular chords in a circle is equivalent to the square on the diameter.

Ex. 912. The hypotenuse of a right triangle is 20, and the projection of one arm upon the hypotenuse is 4. What is its area?

Ex. 913. 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. 914. Transform a given triangle into another triangle containing two given angles.

Ex. 915. Prove geometrically the algebraic formula (a + b) (c + d) : ac+be+ad + bd.

=

Ex. 916. If in any triangle an angle is equal to two thirds of a straight angle (§ 69), then the square on the side opposite is equivalent to the sum of the squares on the other two sides and the rectangle contained by them.

Ex. 917. The two medians RK and SH of the triangle RST intersect at P. Prove that the triangle RPS is equivalent to the quadrilateral HPKT.

Ex. 918. Find the area of a triangle if two of its sides are 6 inches and 7 inches and the included angle is 30°.

Ex. 919. By two different methods find the area of an equilateral triangle whose side is 10 inches.

Ex. 920. The area of an equilateral triangle is 36 √3; find a side and an altitude.

Ex. 921.

By using the formula of Prop. IV, Arg. 8, derive the formula for the area of an equilateral triangle whose side is a.

Ex. 922. What does the formula for T in Prop. IV become if angle C is a right angle?

Ex. 923. Given an equilateral triangle ABC, inscribed in a circle whose center is 0. At the vertex C erect a perpendicular to BC cutting the circumference at D. Draw the radii OD and OC. Prove that the

triangle ODC is equilateral.

Ex. 924. Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles, prove that the bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides.

Ex. 925. A rhombus and a square have equal perimeters, and the altitude of the rhombus is three fourths its side; compare the areas of the two figures.

Ex. 926. The length of a chord is 10 feet, and the greatest perpendicular from the subtending arc to the chord is 2 feet 7 inches. Find the radius of the circle.

Ex. 927. In any right triangle a line from the vertex of the right angle perpendicular to the hypotenuse divides the given triangle into two triangles similar to each other and similar to the given triangle,

Ex. 928. The bases of a trapezoid are 16 feet and 10 feet, respectively, and each of the non-parallel sides is 5 feet. Find the area of the trapezoid. Also find the area of a similar trapezoid, if each of its nonparallel sides is 3 feet.

Ex. 929. A triangle having a base of 8 inches is cut by a line parallel to the base and 6 inches from it. If the base of the smaller triangle thus formed is 5 inches, find the area of the larger triangle.

Ex. 930. If the ratio of similitude of two similar triangles is 7 to 1, how often is the less contained in the greater? HINT. See §§ 418, 503. Ex. 931. Construct a square equivalent to one third of a given square. Ex. 932. If the side of one equilateral triangle is equal to the altitude of another, what is the ratio of their areas?

Ex. 933. Divide a right triangle into two isosceles triangles.