9. Construct a square having each side equal to 2 in. Construct a second square that is double the first. Measure and check by computation.

10. Construct a square having one side 3.2 cm. Construct a second square three times the area of the first. Measure the sides and check by computation.

387. The Steel Square. One of the most useful mechanical instruments is the steel square, or carpenter's square, shown in the figure. It is made in various sizes,

but the most common size is with the longer arm, called the body, blade, or stock, 24 inches in length and 2 inches wide; and with the shorter arm, called the tongue, 16 or 18 inches in length and 1 inches wide. The blade and the tongue form a right angle.

The

Many books have been written on the uses of the steel square. principles involved in using the steel square are mainly those involved in the solution of the right triangle, and in similar triangles (§ 420). One who understands the right triangle and similar triangles can devise many uses for the steel square, and can readily see the principles underlying the various uses of this instrument given in treatises on the steel square.

EXERCISES

1. To find the length of a brace forming the hypotenuse of a right triangle that has a base of 8 ft. and an altitude of 7 ft., one could take 8 in. on the blade and 7 in. on the tongue of a square, as shown in the figure, and then measure the distance AB. This will give the length of the brace to scale. Explain and state theorem involved. 2. Find by means of the carpenter's square the length of a rafter that has a run of 15 ft. and a rise of 10 ft.

3. Show how any number of squares may be added by means of the steel

square. That is, find the side of a square

rafter

run

B

that is equivalent to the sum of any number of squares,

rise

A

B

388. Problem. To transform a given quadrilateral into an

equivalent triangle.

Given the quadrilateral BCDE.

To transform BCDE into an

equivalent triangle.

Construction. Draw the diagonal EC.

meeting DE produced at A, and draw AC.

ABCE+ACDE=AACE+ACDE.

.. AACD=BCDE.

389. Problem. To transform a given polygon into an equivalent triangle.

To transform the polygon into an equivalent triangle. Construction. The diagonal BD is drawn and ABDH is constructed equivalent to ABDC as in § 388.

Similarly AAGE is constructed equivalent to AAFE.

Then quadrilateral GEDH = polygon ABCDEF.

Construct AKDH = quadrilateral GEDH.

..AKDH=polygon ABCDEF.

EXERCISES

§ 388.

1. Transform a square 3 cm. on a side into an equivalent triangle. *2. Draw a pentagon and transform it into an equivalent triangle.

390. Problem. To construct a square equivalent to a given rectangle or parallelogram.

Given the rectangle ABCD.

To construct an equivalent square. Construction. Produce a shorter side as AB of ABCD to D', making AD'AD, and on AD' as a diameter describe a semicircle.

Produce CB to meet the semicircle at G.

Draw AG and construct the square
AEFG.

Then AEFG is the required square.
Proof. Draw ED' and DG.

But

and

E

F

The student should discuss the problem for the parallelogram. 391. Problem. To construct a square equivalent to a given triangle.

392. Problem. To construct a square equivalent to a given polygon.

SUGGESTION. First construct a triangle equivalent to the given polygon and then construct a square equivalent to the triangle.

EXERCISES

CONSTRUCTIONS

1. Construct a square equivalent to a given quadrilateral.

2. Construct a square equivalent to the sum of two triangles.

3. Construct a square containing 15 sq. cm.

SUGGESTION. The square is equivalent to a rectangle 5 cm. by 3 cm. 4. Construct a square equivalent to of a square 7 cm. on an edge.

5. Construct a square equivalent to § of a given square.

6. Construct a square equivalent to the difference in the areas of two given pentagons.

7. Construct a right triangle that is equivalent to a given square.

8. Construct a right triangle that is equivalent to the difference of any two triangles.

9. Construct a square that is equivalent to of a given pentagon. 10. Construct a triangle whose base, altitude, and area are equal to those of a given triangle.

11. Construct the circle with center O in Fig. 1, that is tangent to the semicircle and to the equilateral arch.

SUGGESTION. Find the radius by using rt.^KOE.

12. To inscribe a circle in an equilateral arch formed on AC. (Fig. 2.) SUGGESTION. Construct BD = AC, and DE tangent to the arc.

13. The circle whose center is P is tangent to the three semicircles as shown in Fig. 3. Find its radius, and make the construction as in the figure. Also construct the circle with center O.

SUGGESTION. HP-2a-x. .. (2a−x)2+a2 = (a+x)2.

NOTE. The method of construction in Exercises 11 and 13 shows the use of algebra in analyzing a construction.

14. A hot air pipe that is 10 in. square in cross section changes into a pipe of the same capacity and having its cross section a rectangle 4 in. wide. Find the length of the cross section of the second pipe.

GENERAL EXERCISES

THEOREMS

1. If the diagonals of a quadrilateral intersect at right angles, prove that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other two sides.

2. Prove that two quadrilaterals are equivalent when they have the following parts of one respectively equal to the corresponding parts of the other:

(1) Four sides and one diagonal.

(2) Four sides and one angle.

(3) Two adjacent sides and three angles.

(4) Three sides and two included angles.

3. AB and ECD are two parallel straight lines; BF and DF are drawn parallel to AD and AE respectively. Prove that ▲ ABC and DEF are equal in area.

SUGGESTION. Draw AD, AF, and BD.

4. Prove that the difference of the squares on two straight lines equals the rectangle of which the length is the sum of the straight lines, and the breadth is the difference of the straight lines.

5. Show that the sides of a right triangle may be represented by

if n has such a value that the sides are positive.

6. Show that the sides of a right triangle may be represented by 2n, n2-1, and n2+1. Also by p2+q2, p2−q2, and 2pq.

7. State in integers the sides of four different right triangles.

8. If two sides of a triangle are unequal, the median drawn to the shorter side is greater than the median drawn to the longer side.

9. Prove that the following method of bisecting any quadrilateral by a line drawn from a vertex is correct: Let ABCD be any quadrilateral. Draw AC and BD. Bisect BD at E. Draw EF parallel to AC and meeting BC in F. Draw AF. Then AF bisects the quadrilateral.

SUGGESTION. Draw AE and EC.

10. Inscribe a circle in a triangle ABC, touching its sides at the points D, E, and F. Draw three circles with centers at A, B, and C respectively, each tangent to the other two externally. Show that the points of tangency are D, E, and F.