PLANE GEOMETRY.-BOOK IV. Proposition XXII Problem. Upon a given line to construct a p ar to a given polygon. E D H S ta: Any polygon, as ABCDE, and any line, as FG. quired to construct on FG a polygon similar to ABCDE lution. Draw AC and AD. F and G constructt and Zv equal respectively Ls. roduce the sides from F and G until they meet at H. like manner on FH construct ▲ FHJ, and on FJ, lar respectively to A ACD and ADE. hen, FGHJK is the required polygon. . If one divides two sides of a triangle proportionally and the third side. 2. Lines are in proportion, .If they are segments of two sides of a triangle made by a line he third side. 6. If they are two sides of a triangle and their corresponding s de by a line parallel to the third side. c. If they are two sides of a triangle and the segments of the tl de by the bisector of the angle opposite that side. d. If they are two sides of a triangle and the external segment e. If they are the internal and external segments of a side of a triangle made by the bisectors of an interior and exterior angle at the vertex opposite that side. § 298 f. If they are the altitudes and homologous sides of similar triangles. g. If they are segments of parallel lines made by lines which meet in a point. § 305 § 308 h. If they are homologous sides of similar polygons. $ 299 i. If they are perimeters of similar polygons and any two homologous sides. $311 3. A line is a mean proportional between two other lines, a. If it is the perpendicular to the hypotenuse of a right triangle from the vertex of the right angle, and the other lines are the segments of the hypotenuse. § 312 b. If it is either side about the right angle of a right triangle, and the other lines are the hypotenuse and the segment of it adjacent to that side made by the perpendicular from the vertex of the right angle. § 313 C. If it is the perpendicular to the diameter of a circle from any point in the circumference, and the other lines are the segments of the diameter. §314 d. If it is a tangent to a circle from any point without, and the other lines are a secant from the same point and its external segment. 4. Lines pass through the same point, § 315 a. If they are non-parallel lines that intercept proportional segments upon two parallel lines. § 308 5. Two angles are equal, a. If they are homologous angles of similar polygons. 6. Two triangles are similar, § 299 a. If the angles of one are respectively equal to the angles of the other. $ 300 b. If two angles of the one are respectively equal to two angles of the other. § 301 C. If they are right triangles and an acute angle of one is equal to an acute angle of the other. $ 302 d. If the sides of one are proportional respectively to the sides of the other. § 303 e. If an angle of one is equal to an angle of the other and the including sides are in proportion. f. If their sides are parallel, each to each. § 306 $ 307. g. If their sides are perpendicular, each to each. § 307 h. If they are the corresponding triangles of similar polygons divided by homologous diagonals. § 310 7. Two polygons are similar, a. If they have their homologous angles equal and their homologous sides proportional. § 299 b. If each is composed of the same number of triangles similar each to each and similarly placed. SUPPLEMENTARY EXERCISES § 309 Ex. 433. Construct a triangle whose sides are 6, 8, and 10; then construct a similar triangle whose side homologous to 8 is 5. Ex. 434. Divide a line 10cm long internally in extreme and mean ratio. Ex. 435. The median from the vertex of a triangle bisects every line drawn parallel to the base and terminated by the sides, or the sides produced. Ex. 436. Two circles intersect at A and B, and at A tangents are drawn, one to each circle, to meet the circumference of the other in C and D respectively; BC, BD, and AB are drawn. Prove that BD is a third proportional to BC and AB. Ex. 437. The diameter AB of a circle whose center is O is divided at any point C, and CD is drawn perpendicular to AB, meeting the circumference in D; OD is drawn, and CE perpendicular to OD. Prove that DE is a third proportional to AO and DC. Ex. 438. In the triangle ABC, AD is the median to BC; the angles ADC and ADB are bisected by DE and DF, meeting AC and AB in E and F respectively. Then, FE is parallel to BC. Ex. 439. A secant from a given point without a circle is 1 ft. 6 in. long, and its external segment is 8 in. long. Find the length of a tangent to the circle from the same point. Ex. 440. The radius of a circle is 6 in. What is the length of the tan gents drawn from a point 12 in. from the center? Ex. 441. If the tangent to a circle from a given point is 2m and the radius of the circle is 15dm, find the distance from the point to the circumference. Ex. 442. If from the vertex D of the parallelogram ABCD a straight line is drawn cutting AB at E and CB produced at F, prove that CF is a fourth proportional to AE, AD, and AB, Ex. 443. If the segments of the hypotenuse of a right triangle made by the perpendicular from the vertex of the right angle are 6 in. and 4 ft., find the length of the perpendicular and the length of each of the sides about the right angle. Ex. 444. Find the length of the longest and of the shortest chord that can be drawn through a point 7 in. from the center of a circle whose radius is 19 in. Ex. 445. If the greater segment of a line divided internally in extreme and mean ratio is 36 in., what is the length of the line? Ex. 446. The shorter segment of a line divided externally in extreme and mean ratio is 240dm. Find the length of the greater segment in meters. Ex. 447. Find the shorter segment of a line 12dm long when it is divided internally in extreme and mean ratio. When it is divided externally in extreme and mean ratio. Ex. 448. The tangents to two intersecting circles drawn from any point in their common chord produced are equal. Ex. 449. If the common chord of two intersecting circles is produced, it will bisect their common tangents. Ex. 450. ABC is a straight line, ABD and BCE are triangles on the same side of it, having angle ABD equal to angle CBE and AB: BC= BE: BD. If AE and CD intersect in F, triangle AFC is isosceles. Ex. 451. If in the triangle ABC, CE and BD are drawn perpendicular to the sides AB and AC respectively, these sides are reciprocally proportional to the perpendiculars upon them; that is, AB : AC = BD : CE. Ex. 452. ABCD is a parallelogram. If through O, any point in the diagonal AC, EF and GH are drawn, terminating in AB and DC, and in AD and BC respectively, EH is parallel to GF. Ex. 453. Lines are drawn from a point P to the vertices of the triangle ABC; through D, any point in PA, a line is drawn parallel to AB, meeting PB at E, and through E a line parallel to BC, meeting PC at F. If FD is drawn, triangle DEF is similar to triangle ABC. Ex. 454. If two lines are tangent to a circle at the extremities of a diameter, and from the points of contact secants are drawn terminated respectively by the opposite tangent and intersecting the circumference at the same point, the diameter is a mean proportional between the tangents. Ex. 455. AB and AC are secants of a circle from the common point A, cutting the circumference in D and E respectively. Then, the secants are reciprocally proportional to their external segments; that is, AB : AC = AE: AD. SUGGESTION, Draw CD and BE, and refer to § 322, 6, b. Ex. 456. AB and CD are two chords of a circle intersecting at E. Prove that AE: DECE: BE. Ex. 457. Two secants intersect without a circle. The segments of one are 4 ft. and 20 ft., and the external segment of the second is 16 ft. Find the length of the second secant. Ex. 458. From a point without a circle two secants are drawn, whose external segments are respectively 7dm and 9dm, the internal segment of the latter being 13dm. What is the length of the first secant? Ex. 459. The segments of a chord intersected by another chord are 7 in. and 9 in., and one segment of the latter is 3 in. What is the other segment? Ex. 460. Two secants from the same point without a circle are 24dm and 32dm long. If the external segment of the less is 5dm, what is the external segment of the greater ? Ex. 461. Through a point 7m from the circumference of a circle a secant 28m long is drawn. If the internal segment of this secant is 17m, what is the radius of the circle ? Ex. 462. If from any point in the diameter of a circle produced a tangent is drawn and a perpendicular from the point of contact is let fall on the diameter, the distances from the point without the circle to the foot of the perpendicular, the center of the circle, and the extremities of the diameter are in proportion. SUGGESTION. Draw the radius to the point of contact. Ex. 463. If the sides of a triangle are respectively 1.5Dm, .12Hm, and 10m long, what are the segments into which each side is divided by the bisector of the opposite angle? Ex. 464. If an angle of one triangle is equal to an angle of another, and the perpendiculars from the vertices of the remaining angles to the sides opposite are proportional, the triangles are similar. SUGGESTION. Refer to § 322, 6, c and e. Ex. 465. If two circles are respectively 6 in. and 3 in. in diameter and their centers are 10 in. apart, find the distance from the center of the smaller one to the point of intersection of their common exterior tangent with their line of centers produced. Ex. 466. Two intersecting chords of a circle are 38 ft. and 34 ft. respectively; the segments of the first are 8 ft. and 30 ft. Find the segments of the second. Ex. 467. What is the length of a chord joining the points of contact of the tangents drawn from a point 13 in. from the center of a circle whose radius is 5 in. ? Ex. 468. Chords AB and CD of a circle are produced in the direction of B and D respectively to meet in the point E, and through E the line EF is drawn parallel to AD to meet CB produced in F. mean proportional between FB and FC. Prove that EF is a |