EXERCISES.

1. If a be: ƒ, and c : d =ef, show that a : b = c: d.

bc, show that a: c =

a2: b2.

2. If a : b = 3. A, B, C, D are four points on a straight line, E is any point outside the line. If the parallelograms AEBF, CEDG be completed, show that FG will be parallel to AD.

SIMILAR FIGURES.

307. DEF. Similar figures are those which are mutually equiangular (144), and have their homologous sides proportional.

The homologous sides are those which are adjacent to the equal angles.

(145)

Thus, the figures ABCD, EFGH are similar if

and

ZA ZE, ZB = ZF, ZC = G, etc.,

=

The sides AB and EF are homologous, since they are adjacent to the equal angles A and B, and E and F, respectively; also the sides BC and FG are homologous, etc. The diagonals AC, EG are homologous.

308. The constant ratio of any two homologous sides of similar figures is called the ratio of similitude of the figures.

Proposition 16. Theorem.

309. Triangles which are mutually equiangular are similar.

Hyp. In the As ABC, A'B'C',

let A A', ▲ B = ▲ B',

Proof. Apply the A A'B'C'

to the ▲ ABC so that the pt. A'

coincides with A, and A'B' falls on AB. Let B' fall at

In the same way, by applying the ▲ A'B'C' to the A ABC so that the s at B' and B coincide, we may prove that

AB: A'B' = BC : B'C'.

Combining these two proportions, we have

310. COR. 1. Two triangles are similar when two angles of the one are equal respectively to two angles of the other.

(97)

311. COR. 2. A triangle is similar to any triangle cut off by a line parallel to one of its sides.

312. SCH. In similar triangles the homologous sides lie opposite the equal angles.

Proposition 17. Theorem.

313. Triangles which have their homologous sides proportional are similar.

Hyp. In the As ABC, A'B'C',

AA

Since the first ratio in each of these proportions is the same, the second and third ratios in each are equal respectively; that is,

Since the numerators are the same,

... AE = A'C', and DE = B'C'.

... ▲ A'B'C' = ▲ ADE,

having the three sides equal each to each (108). But A ADE is similar to the ▲ ABC. ... ▲ A'B'C' is similar to the ▲ ABC.

EXERCISE,

Q.E.D.

AB 8,

=

If the sides of the triangle in Prop. 14 are 10, find the lengths of the segments

=

BC 12, and AC =
BD and CD,

Proposition 18. Theorem.

314. Two triangles which have an angle of the one equal to an angle of the other, and the sides about these angles proportional, are similar.

Hyp. In the As ABC, A'B'C', let

But

AA

DE is || to BC,

.. the As ABC, ADE are similar.

.. AB : AD = AC: AE,

C B'

homologous sides of similar ▲s are proportional (307).

A'B' = AD.

.. AB: AD = AC : A'C'.

(311)

(Cons.)

(Нур.)

... A'C' = AE. (Comparing the two proportions)

.. AA'B'C' ▲ ADE,
= Δ

having two sides and the included equal, each to each (104).

... ▲ A'B'C' is similar to the ABC. Q.E.D. 315. SCH. From the definition (307), it is seen that two conditions are necessary that polygons may be similar: (1) they must be mutually equiangular, and (2) their homologous sides must be proportional. In the case of triangles we learn from Props. 16 and 17 that each of these conditions follows from the other; so that one condition is sufficient to establish the similarity of triangles.

This, however, is not necessarily the case with polygons of more than three sides; for even with quadrilaterals, the angles can be changed without altering the sides, or the proportionality of the sides can be changed without altering the angles.

Proposition 19. Theorem.

316. Two triangles which have their sides parallel or perpendicular, each to each, are similar.

Hyp. In the As ABC, A'B'C', let AB, AC, BC be l or L, respectively, to A'B', A'C', B'C'.

To prove ▲ ABC similar to ▲ A'B'C'.

Proof. Since the sides are || or

A

each to each, the included s

B'

are equal or supplementary (80,

83).

... three hypotheses may be made:

(1) A + A′ = 2 rt. ≤ s, B + B′ = 2 rt. ≤ s, C + C′ = 2 rt. ≤ s.

The first and second hypotheses are inadmissible, since the sum of the /s of the two As cannot exceed 4 rt. Zs (97). Therefore the third is the only admissible hypothesis.

... the As ABC, A'B'C' are similar,

being mutually equiangular (309).

Q.E.D.

317. SCH. The homologous sides of the two triangles are either the parallel or the perpendicular sides.

EXERCISE.

If the sides of the triangle in Prop. 15 are AB = 16, BC 10, and AC 8, find the lengths of the segments BD and CD.

=