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Ex. 454. Passing through a point and touching a given circle.
253. No general method can be given for the solution of exercises; a great many, however, can be solved (1) By a gradual putting together of the given parts.
(Remark 233.) (2) By means of an analysis. (3) By means of loci.
MISCELLANEOUS EXERCISES Ex. 457. Through a given point, to draw a line cutting off equal parts on the sides of a given angle,
Ex. 458. Through a given point, to draw a line making a given angle with a given line.
Construct an isosceles triangle, having given :
Ex. 464. The altitude upon the hypotenuse and one of the segments of the hypotenuse.
Ex. 465. The sum of the arms and one acute angle.
Ex. 466. To find a point in one side of a triangle which is equidistant from the other two sides.
Ex. 467. Find the locus of the vertex of a right triangle, having a given hypotenuse.
Ex. 468. From a point P, in the circumference of a circle, to draw a chord, having a given distance from the center.
Ex. 469. In a given circle, to draw a diameter having a given distance from a given point.
Ex. 470. Through a given point without a circle, to draw a secant having a given distance from the center.
Ex. 471. Through two given points in a circumference, to draw two equal parallel chords.
Ex. 472. Trisect a given straight angle.
Ex. 474. Through a given point, to draw a line of given length terminating in two given parallel lines.
Ex. 475. Through a given point, to draw a line making equal angles with two given lines.
*Ex. 476. To bisect an angle formed by two lines, without producing them to their intersection.
To construct a triangle, having given :
Ex. 477. a, ZB, ha.
(Note 230.) Ex. 482. a, ZB, mc. * Ex. 483. ma, mb, hc. * Ex. 484. ma, ho, he.
* Ex. 485. ma, mo, mc.
To construct a square, having given:
Ex. 486. The diagonal.
To construct a rectangle, having given :
Ex. 489. One side and the diagonal.
To construct a rhombus, having given :
Ex. 492. The two diagonals.
To construct a parallelogram, having given :
Ex. 497. Two sides and one altitude.
Ex. 498. Two sides and an angle.
Ex. 501. The diagonals and the angle formed by the diagonals.
254. In the analysis of a problem relating to a trapezoid, draw a line through one vertex, A, either parallel to the opposite arm, DC, or parallel to a diagonal, DB.
To construct a trapezoid, having given :
Ex. 502. The four sides.
Ex. 503. The bases and the base angles.
Ex. 506. One base, the diagonals, and the angle formed by the diagonals.
Ex. 507. To draw a common exterior tangent to two given circles. Ex. 508. To draw a common interior tangent to two given circles.
Ex. 509. About a given circle, to circumscribe a triangle, having given the angles.
Ex. 510. Find the locus of the midpoints of the chords that pass through a given point in the circumference.
Ex. 511. Find the locus of the midpoints of the secants that pass through a given point without a circle.
Ex. 512. In a given circle, to inscribe a triangle, having given the angles.
*Ex. 513. From a given point in a circumference, to draw a chord that is bisected by a given chord.
Ex. 514. Given a point, A, between a circumference and a straight line. Through A, to draw a line terminated by the circumference and the given line, and bisected in A.
Ex. 515. Given two points, A and B, on the same side of a line, CD. To find a point, X, in CD, such that Z AXC = _ BXD.
Ex. 516. The bisectors of the angles of a circumscribed quadrilateral meet in a point.
PROPORTION. SIMILAR POLYGONS
255. A proportion is a statement expressing the equality of two ratios, as = or a:b=c:d.
256. The first and the fourth terms of a proportion are called the extremes, the second and the third, the means.
257. The first and the third terms are called the antecedents, the second and the fourth the consequents.
Thus, in the proportion, a:b=c:d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents.
258. When the means of a proportion are equal, either mean is said to be the mean proportional between the first and the last terms, and the last term is said to be the third proportional to the first and the second terms.
Thus, in the proportion, a:b=b:c, b is the mean proportional between a and c, and c is the third proportional to a and b.
259. The last term is the fourth proportional to the first three. Thus, in the proportion, a:b=c:d, d is the fourth proportional to a, b, and c.
260. A series of equal ratios is called a continued proportion.
261. The two terms of a ratio must be either quantities of the same kind, or the quantities must be represented by their numerical measures only.