245.

PROPOSITION XLIII.

Problem. Upon a given line, homologous to a given side of a given polygon, to construct a polygon similar to the given polygon.

E

Given: The line A' B' and the polygon P.

To Construct-On A' B', homologous to A B, a polygon, P', similar to P.

Sug. Draw the diagonals A C and A D. Under what conditions will ▲ A' B'C' be similar to ▲ A B C? $205. When will P and P' be similar? $205.

SUPPLEMENTARY EXERCISES.

191. To inscribe in a given circle a similar to a given A.

192. To circumscribe about a given circle a ▲ similar to a given A.

BOOK IV.

AREAS.

QUADRILATERALS.

(1) What is meant by a unit of length? a unit of surface? a unit of volume? Give examples of each.

(2) What is meant by a line 5 yards long? a surface containing 25 square feet? a volume of 125 cubic feet? Give other examples.

(3) Illustrate the difference between equal and equivalent figures.

trate.

(1) What is meant by the base of a polygon? Illustrate. (5) What is meant by the altitude of a polygon? Illus

246.

PROPOSITION I.

Draw 2 parallelograms on equal bases and having equal altitudes. Write the steps to show how they are related.

247.

Cor. Using the same figures, can you compare 2 As having equal bases and equal altitudes?

EXERCISES.

193. What is the path of the vertex of a ▲ of constant area on a fixed base?

191. Draw a line from the vertex of a \ to the middle of the opposite side. How does it divide the ? Prove.

Draw lines from vertex to points which are distant,, of the base from either extremity of it. Compare these As.

195. Draw a parallelogram, A B C D. Join B D. Take any point, P, on B D and join with A and C. Compare As DP C and D PA.

=

196. Draw 2 As, A B C and A'B'C', in which B C - B'C' and A CA' C' and C-the supplement of C. Can you prove the As equivalent?

197. Construct ▲ A B C, then on the same base construct a rectangle having twice the area of the A. Prove.

A G the common linear unit of measurement.

Given: The two rectangles B D and B F with equal altitudes, but unequal bases. Compare the rectangles.

Case I. Let the base A D contain the unit A G 8 times, and the base A F contain the unit 3 times. Write an equa

tion expressing the relation of A F to A D.

Call it (1). At

the points of division erect is to B C. Into what is BD divided?

BF? Express the relation between these two rectangles by an equation. Call it (2). Compare (1) and (2).

Given: The rectangle m n and m'n' with equal altitudes? Suppose p n = 1 dm. and p'n'dm., using the mm. as the unit, what is the ratio of these bases? Erect Is at the extremities of each mm. Compare the areas of m n and m' n'. Are the bases commensurable?

In Fig. 1, could the relation between B D and B F be expressed if we used AG as the common unit? Does the relation between B D and B F change if the unit is halved? What effect does it have on the small rectangles when we decrease the size of the common unit? Could we express the relation between B D and B F if we took A G as a common unit? How small a fractional part may we take and still express the true relation ?

Suppose we think of the common unit as 1 millionth of A G, how many rectangles may be formed in B F? in B D? in E D?

[Review on Principles of Limits.—(Review § 195.)

Let AM and A'M' be two equal variables which constantly* A B and A' B' respectively. Let us compare A B and A' B'. If possible, suppose A B > A'B', measure off on A B a distance A CA' B'. What is the limit A M approaches?

(* means approach as a limit.)

What is the limit A' M' ap

=

May the limit of A M pass A C? proaches? Can A'M' ever reach A' B' or A C? What is the relation of A M and A' M? Do you see any absurdity in supposing A B A'B'? Now suppose that A B < A'B' and let A'B' be measured off on A B produced and let A D A'B'. What will A M? A'M'? Can A'M' become greater than A B? Can A M become equal to A B? How are the variables A M and A'M' related? What absurdity by our last supposition? Now if A B cannot be greater than A'B' and it cannot be less than A' B', what condition must exist? Write a general statement of what we have proved and memorize it.]

Let D C and D F in the 2 □s AC and A F be incommensurable, but having same altitude. Suppose a unit of length, u, is contained an exact number of times in D C, say 3 times, and in DG once, with remainder. What do we say of DC and DG? Erect a 1 at G forming the □ A G. AC related? Now imagine the unit of crease. How does it affect D G? GF? What does D G ?

How are A G and comparison to de

What does AG? Now do you see that

AG DG

A C

DC? Do

you see that each member of the equation is a variable? Does each approach a limit? Are they always equal? What have we learned about two equal variables as they approach limits? Draw conclusion. Call it Prop. II.

249.

Which side of a rectangle may be considered the base? Can A D be considered the base of each of the rectangles,