NUMERICAL EXERCISES

1. An inscribed angle intercepts an arc of 40° (50°, 60°, 75°). How many degrees in the angle?

2. An angle of 20° (30°, 45°, 50°, 67° 30′) is inscribed in a circle. How many degrees in the intercepted arc ?

3. The sides of an inscribed triangle subtend arcs of 100°, 120°, and 140°. How many degrees in each angle of the triangle?

4. The arcs subtended by the sides of an inscribed triangle are in the ratio 1:2:3. What kind of triangle is it?

5. The sides of an inscribed quadrilateral subtend arcs in the ratio 1:2:3:4 (3:5:7:9). How many degrees in each angle of the quadrilateral?

6. In the preceding exercise how many degrees in the angles between the diagonals of the quadrilateral?

7. A triangle is inscribed in a circle, and another triangle is circumscribed by drawing tangents at the vertices of the inscribed triangle. The angles of the inscribed triangle are 40°, 60°, and 80°. Find all the other angles of the figure.

8. The vertex angle of an inscribed isosceles triangle is 100°. How many degrees in the arcs subtended by each of the sides?

9. The bases of an inscribed isosceles trapezoid subtend arcs of 100° and 120°. How many degrees in each angle of the trapezoid (a) if the bases are on the same side of the center? (b) if they are on opposite sides of the center?

10. At the vertices of an inscribed quadrilateral tangents are drawn to the circle, forming a circumscribed quadrilateral. The arcs subtended by the sides of the inscribed quadrilateral are in the ratio 3:4:5:8. Find

(a) the angles of each quadrilateral;

(b) the angles between the diagonals of the inscribed quadrilateral;

(c) the angles between the opposite sides of the inscribed quadrilateral produced to intersect;

(d) the angles between the sides of the inscribed and those of the circumscribed quadrilateral.

11. A regular hexagon is inscribed in a circle and all its diagonals are drawn. How many degrees in each angle of the figure?

12. The angle formed by two tangents drawn to the same circle is 100°. How many degrees in the two arcs subtended by the chord of contact?

13. If the angle between two tangents to the same circle is 60°, what kind of triangle is formed when the chord of contact is drawn?

14. Given two tangents to a circle; the major arc contains 200°. How many degrees in the angle formed by the tangents?

15. The sides of an inscribed pentagon subtend arcs each of which is 10° greater than the preceding one. Draw the diagonals of the pentagon and determine each angle of the figure.

16. In the preceding exercise draw tangents at the vertices of the inscribed pentagon and determine each of the interior angles of the resulting circumscribed pentagon.

17. The diagram shows how the latitude of a place may be determined by observation of the pole star. Let EE' represent the equator, AA' the axis of the earth, P the place whose latitude is to be found, PF a plane tangent to the earth at P (the horizon), and PS the line E of observation of the pole star. Then Za' represents the latitude, and ∠a is called the altitude of the pole star. For practical purposes we may assume that AA' || PS.

Prove that Za' = 2a.

P

S

F

A

a

a'

E'

A'

18. An inscribed angle is formed by the side of a regular inscribed hexagon and the side of a regular inscribed decagon. How many degrees in the angle? (Give two solutions.)

19. Solve Ex. 18 if the inscribed angle is formed by sides of the following regular polygons:

(a) triangle and square;

(b) pentagon and octagon;

(c) hexagon and octagon;
(d) pentagon and dodecagon.

LOCI

294. The definition of a circle may be used to introduce a very important geometric idea. All points in a plane which are two inches from a given point O lie in a circle whose center is the given point O and whose radius is two inches long.

Conversely, every point in this circle is two inches from 0.

0.

These two statements are sometimes replaced by the one statement that the circle about O is the locus (that is, place) of the points two inches from O. In general,

The locus of a point satisfying a given condition is the figure containing all the points that fulfill the given condition (or answer the given description), and no other points. Hence the following

295. Rule for solving Locus Problems:

1. Locate a number of points which satisfy the given condition, and thus obtain a notion of what the locus is.

2. Prove that every point satisfying the given condition lies in the assumed locus.

3. Prove that every point of the assumed locus satisfies the given condition.

In the following simple exercises, however, the answers may be stated without proof.

EXERCISES

1. Where are all the houses that are 1 mi. from your school building?

2. Where are all the villages that are 10 mi. from your own town? 3. Sound travels about 1100 ft. per second. If a cannon is fired from a certain point, what is the locus of all persons who hear the report after 3 sec.?

4. A ladder leans against a wall. A man stands on the middle rung. If the ladder slips down, what is the locus of the man's feet? 5. What locus problems are suggested to you by the opening of a book or a door? by a seesaw? a pendulum? the governor of an engine? a clock?

Theorem I. The locus of a point X at a given distance'd from a given point P is a circle having Pas its center and d as its radius.

EXERCISES

1. What is the locus of a house 100 ft. from a straight road?

2. On a city map what is the locus of a house 200 ft. from the "mile circle"?

3. Where are all the points that are 2 in. from the surface of the table?

4. What is the locus of the foot of a tree which is 10 ft. from the wall of a round tower?

5. What is the locus of a point 1 in. from the surface of a spherical shell?

Theorem II. The locus of a point X at a given distance d from a given line I consists of two lines parallel to l, one on each side of l, and d units from it (§ 237).

Theorem III. The locus of a point X at a given distance d from a given circle whose radius is R consists of two circles concentric with the given circle and with

R

dd

radii R+ d and R d respectively (§ 269).

EXERCISES

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1. A motor boat sails up a straight canal midway between the banks. What is the locus of the boat?

2. The hands of a clock describe concentric circles. What is the locus of a point equidistant from these circles?

3. What is the locus of a point equidistant from two opposite walls of a rectangular room?

Theorem IV. The locus of a point equidistant from two parallel lines is a parallel line midway between the two parallels (§ 237).

Theorem V. The locus of a point equidistant from two concentric circles is a concentric circle midway between those two circles (§ 269).

What is the radius of the locus if the radii of the given circles are r1 and r2?

EXERCISES

1. The poles of a telephone line are to be each equidistant from two houses. What is the locus of these poles?

2. What is the answer in Ex. 1, if the poles of the telephone line are to be equidistant from two intersecting straight roads?

Theorem VI. The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the two points (§ 239).

Theorem VII. The locus of a point equidistant from two intersecting straight lines consists of the pair of straight lines that bisect the angles formed by the two given lines (§ 240).

EXERCISES

1. In front of a rectangular house, trees are planted in such a manner that the lines joining the foot of each tree to the two nearest corners of the house inclose a rightangle. Where are the trees situated?

2. How would the trees be located, if the lines joining the foot of each tree to the two nearest corners of the house inclosed an angle of 40°?

3. Given two fixed points A and B. Through A a line is drawn perpendicular to a line passing through B. Call their point of intersection C. What is the locus of C?

4. State the locus theorems underlying Exs. 1-3 (§ 293).

296. Intersection of loci. A point sometimes fulfills more than one condition. The solution then consists in finding the points common to two or more loci.