Ex. 39. The sides of a triangle are 8, 10, 12.

Find the areas of the triangles made by the bisector of the angle opposite the side 12.

Ex. 40. In a circle of area 275 sq. ft., a rectangle of area 150 sq. ft. is inscribed. Show how to find the sides of the rectangle.

EXERCISES. GROUP 63

EXERCISES INVOLVING THE METRIC SYSTEM

Ex. 1. Find the area of a triangle of which the base is 16 dm. and the altitude 80 cm.

Ex. 2. Find the area of a triangle whose sides are 6 m., 70 dm., 800 cm.

Ex. 3. Find the area in square meters of 2 circle whose radius is 14 dm.

Ex. 4. If the hypotenuse of a right triangle is 17 dm. and one leg is 150 cm., find the other leg and the area.

Ex. 5. If the circumference of a circle is 1 m., find the area of the circle in square decimeters.

Ex. 6. Find the area in hectares, and also in acres, of a circle whose radius is 100 m.

Ex. 7. If the diagonal of a rectangle is 35 dm. and one side is 800 mm., find the area in square meters, and also in square inches.

Ex. 8. Find the area of a trapezoid whose bases are 600 cm. and 2 m., and whose altitude is 80 dm.

Ex. 9. If a rectangular field is 700 dm. long and 200 m. wide, find its area in hectares and in acres.

Ex. 10. In a given circle two chords, whose lengths are 15 dm. and 13 dm., intersect. If the segments of the first chord are 12 dm. and 3 dm., find the segments of the second chord.

Ex. 11. Find in decimeters the radius of a circle equivalent to a square whose side is 1 ft. 6 in.

Ex. 12. Find in feet the diameter of a wheel which, in going 10 kilometers, makes 5,000 revolutions.

BOOK VI

LINES, PLANES AND ANGLES IN SPACE

DEFINITIONS AND FIRST PRINCIPLES

497. Solid Geometry treats of the properties of space of three dimensions.

Many of the properties of space of three dimensions are determined by use of the plane and of the properties of plane figures already obtained in Plane Geometry.

498. A plane is a surface such that, if any two points in it be joined by a straight line, the line lies wholly in the surface.

499. A plane is determined by given points or lines, if no other plane can pass through the given points or lines without coinciding with the given plane.

A

500. Fundamental property of a plane in space. plane is determined by any three points not in a straight line.

The importance of the above principle is seen from the fact that it reduces an unlimited surface to three points, thus making a vast economy to the attention. It also enables us to connect different planes, and treat of their properties systematically.

(319)

501. Other modes of determining a plane. A plane may also be determined by any equivalent of three points not in a straight line, as by

a straight line and a point outside the line; or by

two intersecting straight lines; or by

two parallel straight lines.

It is often more convenient to use one of these latter methods of determining a plane than to reduce the data to three points and use Art. 500.

502. Representation of a plane in geometric figures. In reasoning concerning the plane, it is often an advantage to have the plane represented in all directions. Hence, in drawing a geometric figure, a plane is usually represented to the eye by a small parallelogram.

This is virtually a double use of two intersecting lines, or of two parallel lines, to determine a plane (Art. 501).

503. Postulate of Solid Geometry. The principle of Art. 499 may also be stated as a postulate, thus:

Through any three points not in a straight line (or their equivalent) a plane may be passed.

504. The foot of a line is the point in which the line intersects a given plane.

505. A straight line perpendicular to a plane is a line perpendicular to every line in the plane drawn through its foot.

A straight line perpendicular to a plane is sometimes called a normal to the plane.

506. A parallel straight line and plane are a line and plane which cannot meet, however far they be produced.

507. Parallel planes are planes which cannot meet, however far they be produced.

508. Properties of planes inferred immediately.

1. A straight line, not in a given plane, can intersect the given plane in but one point.

For, if the line intersect the given plane in two or more points, by definition of a plane, the line must lie in the plane.

Art. 498.

2. The intersection of two planes is a straight line.

For, if two points common to the two planes be joined by a straight line, this line lies in each plane (Art. 498); and no other point can be common to the two planes, for, through a straight line and a point outside of it only one plane can be passed.

Art. 501.

Ex. 1. Give an example of a plane surface; of a curved surface; of a surface, part plane and part curved; of a surface composed of different plane surfaces.

Ex. 2. Four points, not all in the same plane determine how many different planes ? how many different straight lines?

Ex. 3. Three parallel straight lines, not in the same plane, determine how many different planes ?

Ex. 4. Four parallel straight lines can determine how many different planes ?

Ex. 5. Two intersecting straight lines and a point, not in their plane, determine how many different planes ?

PROPOSITION I. THEOREM

509. If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of those lines.

A

M

B

D

N·

Given AB 1 lines BC and BD, and the plane MN passing through BC and BD.

Proof. Through B draw BG, any other line in the plane MN.

Draw any convenient line CD intersecting BC, BG and BD in the points C, G and D, respectively.

Produce the line AB to F, making BF= AB.

Connect the points C, G, D with A, and also with F.

Then, in the A ACD and FCD, CD=CD.

ACCF, and AD=DF.

Ident.

Art. 112.

:. A ACD= ▲ FCD.

(Why?)

:: 4 ACD= / FCD.

(Why?)

Then, in the ▲ ACG and FCG, CG=CG,

(Why?)

AC

CF, and Z ACG = LFCG.

(Why ?)

:: A ACG= ▲ FCG.

(Why ?)