GEOMETRICAL PROBLEMS.

SECTION V.

PROBLEM I.

To draw a line parallel to agiven straight line, at a given distance.

A

CD

B

At the given distance C D from the line A B describe an arc of a circle; at the same distance from another part of the same line, describe a similar arc; draw a

straight line touching these arcs without cutting them, and it will be parallel to A B.

PROBLEM II.

To draw a perpendicular from any part of a given

line.

F

E

-B

D

From the point C in the given line A B, describe an arc of a circle greater than a quadrant. From D to F, at the distance of the radius set off a part of the arc from D to E and from E to F; from the points E and F describe intersecting arcs at G; draw a line from C to G and it will be perpendicular to A B.

PROBLEM III.

To bisect or divide a given line into two equal parts.

*

A

-B

Ж

At any distance greater than half of the given line A B with the points A B as centres, describe intersecting arcs above and below the given line; draw a straigh line between the intersecting arcs, and it will divide the line A B into two equal parts.

7

PROBLEM IV.

To divide a given straight line into any number of equal parts.

Divide the line A B into seven equal parts.

A

B

D

From the points A, B, draw the lines A D and B C, indefinitely, parallel to each other. On them, from A and B set off the number of parts required: join the opposite points, and it will divide the line A B as proposed.

PROBLEM V.

To divide a line in any given proportion.
Divide A B in the proportion of 3 to 5.

From the point A, draw A D of indefinite length. On it, set off eight equal divisions of indefinite length : join B with the eighth division, and parallel to it join C to the fifth division. Then A B will be divided in the proportion required.

PROBLEM VI.

To find a mean proportional to two given lines. Find a mean proportional between the lines A B and B C.

Draw a line A C equal to A B and B C. From the centre of the line A C as diameter, describe a semicircle. At the point B draw a perpendicular B D and it will be a mean proportional between A B and B C.

PROBLEM VII.

To find a third proportional to the less of two given lines.

Find a third proportional to the less of the lines C B

Draw an acute angle having the sides equal to C B, CD; from C D cut off C A equal to C B; join B D and draw A E parallel to it, then CE will be the third less proportional to C D, C B.