B

NOTE.-Angle ABC is a right angle, and the lines drawn from A and C to any point in the arc of the semicircle would form a right angle (Euc. III., 31).

Problem 38.

To construct a rhombus, its side AB and one diagonal AC being given.

1. With centres A and C, and radius AB, describe arcs cutting at D and B.

2. Join AD, DC, CB, and BA. Then ADCB is the rhombus required.

Problem 39.

To construct a rhombus, its side AB and an angle C being given.

1. Make angle BAD equal to C (Pr. 10), and cut off AE equal to AB.

2. With centres E and B, and radius AB, describe arce intercepting in the point F.

3. Join EF, FB. Then AEFB is the required rhombus.

Problem 40.

To construct a rhomboid, its two adjacent sides AB and BC, and a diagonal AC, being given.

B

B

1. From A in AB, with the diagonal AC as radius, describe

an arc.

2. From B, with radius BC, cut the arc in C.
3. From C, with AB as radius, describe an arc.
4. From A, with radius BC, cut the arc in D.

5. Join BC, CD, and DA. Then ABCD is the rhomboid required.

Problem 41.

To construct a rhomboid, its two adjacent sides AB and BC, and angle D being given.

1. At A in AB make the angle BAF equal to the given angle D (Pr. 10), and cut off AE equal to BC.

2. From B, with radius BC, describe an arc above B. ·
3. From E as centre, with radius AB, cut the arc in C.
4. Join EC and CB. Then AECB is the required rhomboid.

Problem 42.

To construct a trapezium equal to a given trapezium ABCD.

1. Make line EF equal to CD, and the angle at £ equal to the angle at D (Pr. 10).

3. From the point G, with radius AB, and from F, with radius BC, describe arcs cutting in H.

4. Join GH, FH. Then the trapezium EFGH shall be equal to the given trapezium ABCD.

Problem 43.

To construct a trapezium, having its adjacent pairs of sides equal respectively to two given lines AB and CD, and its diagonal equal to the given line EF.

F

A

H

1. From centre E, with AB radius, and from centre F, with CD radius, describe arcs cutting in G and H.

2. Join EG, GF, FH, and HE.

trapezium.

Then EGFH is the required

Problem 44.

To construct a trapezium when the length of the diagonal AB, and the angles at its extremities A and B are given.

1. Make any straight line CD equal to the diagonal AB.

E

B

2. Make angles at C'equal and correspondent to the angles at A; and angles at D equal and correspondent to the angles at B (Pr. 10).

3. Produce their sides until they meet. Then the figure CEDF is the required trapezium.