"Scholium to Proposition I.

"If the point B is on uneven ground, measure off any distance BF, draw FG perpendicular to AC and operate from G. "Subtract BF from H to get point C."

NOTE: A norma is another instrument used by the ancients to draw perpendicular lines. It consists of three rods placed at right angles to each other.

Note the lack of perspective in the drawing.

If the angles BDA and BDC are equal, what kind of a triangle is ADC?

Why does BC equal AD?

Bettinus gave a proof for this proposition by proving the triangles ABD and CBD equal.

POLYGONS

A. TRAPEZOIDS

1. (a) Roads and streets do not always run at right angles to each other, but lot lines are usually perpendicular to the street on which the lots face. In such a city block most of the lots are rectangular, but a few will be in the shape of triangles and trapezoids.

(b) A trapezoid is a figure inclosed by four straight lines, only two of which are parallel.

(c) The two parallel sides are the bases. The two non-parallel sides are the legs.

(d) If the two legs are equal, the figure is an isosceles trapezoid.

2. (a) What is the shape of each lot in the above plot? (b) In order to find the area of lots 6 and 12, one must know how to find the area of a trapezoid. (c) The symbol for trapezoid is.

3. (a) Draw a trapezoid ABCD and its altitude h. (b) Draw one diagonal as BD.

=

ABCD.

(c) Into what two parts does BD divide the trapezoid?
(d) We see that ▲ 1+ ▲ 2
(e) Let b1 (read b sub 1)
lower base of the

=

;

the base of ▲ 1 or the

and b2 (read b sub 2) the base of A 2 or the upper base of the □.

=

(f) What is the height of each A? The height of the trapezoid is a line perpendicular to the bases.

(g) Translate this formula into an English statement.

10 cm., b2

and h

=

3 cm., find S.

5. The formula for the area of a trapezoid may be found

(a) Draw two trapezoids that are exactly equal.
(b) Cut out one and place it beside the other as in
the illustration.

(c) What kind of a figure is the result?

(d) What is the base of the new figure? The height?
(e) What is its area?

(f) What part of the new figure is the trapezoid?
(g) Therefore, what is the area of the trapezoid?
6. (a) Find the areas of the following trapezoids.
(b) Use the formula

7. In the diagram page 161 measure lots 1, 6, 7, 8, and 12 to the nearest tenth of a centimeter.

areas if 1 cm. equals 60 ft.

Find their

8. (a) A room has a bay window and is shaped like the

diagram.

24

(b) The length of the room is

24 feet.

(c) Find the scale and the entire area of the room

including the bay window.

9. (a) Draw a trapezoid with a

very small upper base.

What other figure does it resemble?

(b) Show that the formula for the area of a trapezoid may be used for the triangle by considering the upper base 0.

10. (a) To measure an irregular shaped piece of land, as that in the bend of a river, a straight line may be run so as to cut off the bend, as AB. At equal intervals on AB perpendiculars are run. These divide the land into approximate trapezoids.

A

X

B

(b) This figure is drawn to a scale of 1" to 40'.
Find the length of parts AB and of the perpen-

diculars to AB.

(c) Find the area of each near trapezoid.