Ex. 80. Find the angle inscribed in a semicircle the sum of whose sides is a maximum.

Ex. 81. The bases of a trapezoid are 160 and 120, and the altitude 140. Find the dimensions of two equivalent trapezoids into which the given trapezoid is divided by a line parallel to the base.

Ex. 82. If the diameter of a given circle be divided into any two segments, and a semicircumference be described on the two segments on opposite sides of the diameter, the area of the circle will be divided by the semicircumferences thus drawn into two parts having the same ratio as the segments of the diameter.

Ex. 83. On a given straight line, AB, two segments of circles are drawn, APB and AQB. The angles QAP and QBP are bisected by lines meeting in R. Prove that the angle R is a constant, wherever P and I may be on their arcs.

Ex 84. On the side AB of the triangle ABC, as diameter, a circle is described. EF is a diameter parallel to BC. Show that EB bisects the angle ABC.

Ex. 85. Construct a trapezoid, given the bases, one diagonal, and an angle included by the diagonals.

Ex. 86. If, through any point in the common chord of two intersecting circles, two chords be drawn, one in each circle, through the four extremities of the two chords a circumference may be passed.

Ex. 87. From a given point as center describe a circle cutting a given straight line in two points, so that the product of the distances of the points from a given point in the line may equal the square of a given line segment.

Ex. 88. AB is any chord in a given circle, P any point on the circumference, PM is perpendicular to AB and is produced to meet the circle at Q; AN is drawn perpendicular to the tangent at P. Prove the triangles NAM and PAQ similar.

Ex. 89. If two circles ABCD and EBCF intersect in B and C and have common exterior tangents AE and DF cut by BC produced at G and H, then GÃ2=BC2+AE2,

PRACTICAL APPLICATIONS OF PLANE GEOMETRY

EXERCISES.

GROUP 88

(BOOK I)

1. Take a piece of paper having a straight edge and fold the paper so as to form a right angle. What geometrical principle or definition have you used?

2. By use of a board with a straight edge, test the accuracy of the outside of a carpenter's square by a method indicated in the diagram. How, then, would you test the accuracy of the inside angle of the square? What geometric principle have you

used in each case?

3. In Ex. 2 prove that the error in the outside angle of the carpenter's square, if there be any, equals one-half the angle, x, between the outside lines of the square as shown in the diagram (denote the error by e and show that e + e = x).

This principle is important because it is essentially the method used in correcting the axis of a telescope, and hence in correcting instruments of which the telescope is a part, as various surveying and astronomical instruments.

What

4. By use of a carpenter's square and a given straight edge, lay off a series of parallel lines. property of parallel lines have you used?

5. Tell how to construct a carpenter's miter box. 6. The diagram shows a drawing instrument called the parallel rulers. The dotted outline shows the rest of the instrument in another position, the part RS remaining fixed. The distance PQ RS, PR = QS. Hence show that PQ is parallel to RS.

In like manner show that P'Q' is parallel to RS. Hence show PQ is parallel to P'Q'.

If a line be drawn perpendicular to PQ and

another perpendicular to P'Q', the perpendiculars thus drawn will be parallel to each other (Art. 122, lines perpendicular to parallel lines are parallel).

340 COPYRIGHT, 1911

BY CHARLES E. MERRILL COMPANY

The above are cases of linked motion, a kind of mechanism of wide importance. Observe for instance the system of links which connect the driving wheels of a locomotive with the piston in the cylinder, and also the jointed rods connecting the walking beam of a steamboat with the engine. Look up also, in the Century Dictionary, for instance, the words linkage, cell, and parallel motion.

7. The distance between two accessible places separated by an impassable barrier (as between two houses separated by

a pond) may be found by the following method when A no instrument for measuring angles is at hand.

Let A and B be the two places. Take a convenient station C and measure AC and BC. Produce AC to F, making CF = AC. Produce BC to D, making Prove AB

DC

=

BC. Measure DF.

B

D4

F

= DF.

8. In the trusses of steel bridges, why are the beams and rods arranged so as to form a network of triangles as far as possible, and not of quadrilaterals, or pentagons, for instance? (See Ex. 3, p. 76; also Art. 101.)

How is this principle also made use of in forming the frame of a wooden house, or box car, or to strengthen a weak frame or fence of any kind? 9. Draw a map for the following survey notes to the scale of

of survey notes similar to those in Ex. 9 and make a drawing for them. 11. Let DC and FC (p. 342) be two walls perpendicular to the plane Let small mirrors be attached to these walls at A and B. 45°. Let a ray of light pass through Q to A, be reflected to B, and thence to P. Prove that APB is a right angle.

of the paper.

Let C

=

[SUG. The law of reflected light is that the angle of incidence equals the angle of reflection, or, in the figure, x = x and y

=

=

the triangle A B C, x + y + 45° 180°, or x + y points A and B, 2 x + a + 2 y + b = 360° .. a + b

y. Then in 135°. About the

The preceding is the principle of an instrument called the "optical square" used by foresters in constructing right angles. For a ray of light coming from R through a small hole at B above the mirror will make a right angle with the ray coming from Q through P to A.

a

45

12. The velocity of light is determined by the use of a rotating mirror in the following manner: Let AB1 be a mirror plane of the paper and rotating about O as a pivot; let OP, be A1В1. Let LO be a ray of light striking the mirror at O and reflected through M to a small stationary mirror some miles distant, whence it is reflected back through M to 0. On the return of the ray to 0, AB1 will have rotated through a small angle, a, to the position A2B2; A hence the ray will be reflected in the direction OR. Let the pupil show that La - LOR. Since LOR may readily be measured, a is known, and if the rate at which the mirror is rotating is known, the time occupied by the

=

a

B2

A2

B1

a

M

x-a

P1 P

ray in traveling from O to the s ationary mirror and back is determined.

If

LOR

=

2° 19′, OM = 3 mi., and the mirror AB rotates 100 times per second, determine the velocity of light per second.

13. To prove that the image of a point in a plane mirror is on a perpendicular from the point to the mirror and as far behind the

mirror as the object is in front of it, let MM' be the mirror and P the point, and P PAE and PA'E' two reflected rays. Show that

14. From the above show the direction in which a billiard ball must be sent in order to strike a certain point on the table after striking one side of the table.

15. If two mirrors OM and OM' be perpendicular to each other

M

make a construction to show where the point P will appear to be to an eye at E, after the light from P has been reflected from both mirrors.

=

[SUG. Prove BP" = BA + AP' BA + AP.]

16. What would be the application of Ex. 15 to a ball struck on a billiard table?

17. The path of a ray of light before and after entering a glass prism is given by the lines AB and CD. The entire angle by which a ray of light is deflected on passing through the prism is denoted by x. Prove that x = i + r− P. (Zs PBy and PCy are rt. s.)

[SUG. By use of a quadrilateral BPCy, y = 180° - P. Then use the quadrilateral all of whose sides are dotted lines.]

1. Given a fragment of broken wheel show how to find the radius of the wheel.

2. By use of the carpenter's square find the center of a given circle.

3. Show how a pattern maker by the use of a carpenter's square can determine whether the cavity made in the edge of a board or piece of metal is a semicircle.

4. Bisect a given angle by the use of a carpenter's square.

5. By use of squared paper divide a line 11⁄2 inches long into five equal parts. Into 7 equal parts. Into 3 equal parts.

this division on paper ruled in only one direction?

6. Make up and work a similar example for yourself.

Can you make