298. COR. If from any point A in a circumference, a perpendicular be drawn to a diameter BC, and chords AB, AC, be drawn, B since BAC is a right angle, DC: DADA: DB, BC: BA=BA: BD, and BC:CA=CA: CD. Hence 1°. The perpendicular from any point of a circumfer ence to a diameter is a mean proportional between the segments. 2°. The chord drawn from the point to either extremity of the diameter is a mean proportional between the diameter and the adjacent segment. 299. DEFINITION. Two ratios are said to be mutually inverse or reciprocal when the antecedent and the consequent of the one are, respectively, the consequent and the antecedent of the other. Thus B: A is the inverse of 4 : B, and 11: 7 is the inverse of 7: 11. 300. DEFINITION. If four quantities, A, B, C, D, are so related that A: CD: B, or Α D i.e., if the first has to the third the inverse ratio of the second to the fourth, the quantities are said to be inversely proportional, while, as we know, if the first is to the third as the second to the fourth, the quantities are directly proportional. As will be seen in the next proposition, two lines have their segments inversely proportional if a segment of the first is to a segment of the second as the remaining segment of the second is to the remaining segment of the first; while the segments would be directly proportional if a segment of the first were to a segment of the second as the remaining segment of the first to the remaining segment of the second. PROPOSITION XXI. THEOREM. 301. If two chords intersect, their segments are inversely proportional. B A Given: In circle ADB, chords AB, CD, intersecting in 0; OA: OD OC: OB. SCHOLIUM. In deducing such proportions between pairs of sides in similar triangles, the student may find it useful to remember that, as the homologous sides are opposite equal angles, the greater side of the one is to the greater side of the other as the lesser side of the one is to the lesser side of the other. EXERCISE 363. In the diagram for Prop. XI., if a parallel to BC cut AB, AC, and AD, in G, H, and L, respectively, show that AD is divided by GL so that AL: AD = GH: BC. 364. In the diagram for Prop. XII., show that the triangles A'AD and OA'B' are similar to each other and to triangle OAB. 365. In the same diagram, show that the perimeter of triangle OAB is to that of triangle OA'B' as OB is to OB'. 366. In the diagram for Prop. XX., if ▲ B: 2 C = 3: 5, how many degrees in each of the acute angles at A? 367. In the same diagram, if the numerical measures of BC and DC are 10 and 2, respectively, what is the numerical measure of AD? 368. Find also the numerical measures of AB and AC. PROPOSITION XXII. THEOREM. 302. If two secants be drawn from a point without a circle, these secants and their external segments are inversely proportional. Given: Two secants OA, OD, cutting circle ADB in B, A, Join AC and BD. Then since AD, and Zo is common to both, and (264) AAOC is similar to A DOB, (287) .. OA: OD = OC: OB. Q.E.D. (289) EXERCISE 369. Prove Prop. XXI. by joining AD and BC, and showing that OA: OD OC: OB. = 370. In the diagram for Prop. XXII., show that the triangles whose vertices are at the intersection of AC and BD are similar. 371. If two equal chords intersect, their segments are severally equal. 372. If equal chords be produced to meet, the secants thus formed and their external segments will be severally equal. 373. In the diagram for Prop. XXI., show that if the chords are equally distant from the center, they are directly as well as inversely proportional. 374. In the diagram for Prop. XXII., show that if the secants are equally distant from the center, they are directly as well as inversely proportional. 375. If two circles intersect, tangents drawn to them from any point in their common chord produced, will be equal. PROPOSITION XXIII. THEOREM. 303. If a secant and a tangent be drawn from a point without a circle, the tangent is a mean proportional between the secant and its external segment. B Given: A tangent OC touching circle ABC in C, and a secant OA cutting ABC in B, A; To Prove: OA: OC OC: OB. Join AC and BC. Then A and OCB are each meas. by arc BC, (264, 269) ando is common to AOAC and OBC, .. AOAC is similar to A OBC, .. OA: OC OC: OB. (287) Q.E.D. (289) EXERCISES. QUESTIONS. 376. In the diagram for Prop. VIII., if DB: AD=3:7, and AC 55, what is the value of AE?* = 377. If in the preceding question we substitute VII for 3, what is the value of AE? *Value, here and elsewhere, stands for numerical measure, the unit being left undetermined. If decimals occur in a result, it will be sufficient to have two places correct. 378. In the diagram for Prop. X., if AB, AC, BC, = 9, 7, 12, respectively, what is the value of BD and of DC? 379. In the diagram for Prop. XI., if AB, AC, BD, = 9, 7, 20, respectively, what is the value of BC? 380. If the angles at the base of ▲ ABC in Prop XI. are equal, how is the proposition modified? 381. If two triangles have an angle of the one equal to an angle of the other, and the sides about another angle proportional, are they necessarily similar? 382. In the diagram for Prop. XII., if AB, AC, A'B,'= a, b, a', respectively, what is the value of A'C' ? 383. In the diagram for Prop. XVII., if AD, DE, AC, A'D', = 4, 3, 5, 3.2, respectively, what are the values of D'E', A'C' ? 384. In the diagram for Prop. XX., if AB, AC, BC, respectively, what are the values of AD, BD, DC? = 4, 3, 5, 1 of AD, what part m 385. In the diagram for Prop. XXI., if A0 = 1⁄2 OD, and OB what is the value of OC? 386. In the diagram for Prop. IX., if BD is is CG of AG? = 387. In the diagram for Prop. X., if AB: AC, 10: 7, what is the ratio of AD to EC? 388. In the same diagram, if AB = AC, how many degrees are there in angle BCE? 389. In the diagram for Prop. XI., if EC: AD, = 2 : 3, what is the ratio of AB to AC? 390. In the same diagram, if AB is equal to AC, where will the point E fall? 391. In the diagram for Prop. XII., if AC = 12 and OB: OB' = 8:5, what is the value of AD + EC? 392. In the same diagram, if OB bisects angle AOC, AB = 10, and BC 8, what is the ratio of OA' to OC'? = 393. In the diagram for Prop. XIII., if BC : B'C' = m : n, what is the ratio of AB AC to A'B' — A'C' ? 394. In the diagram for Prop. XVII., if AB = 18, and A'B' what is the ratio of AC to A'C'? 395. In the diagram for Prop. XX., if BD: DA = m : n, what is the ratio of the perimeter of triangle ADB to the perimeter of triangle ADC? |