MISCELLANEOUS EXERCISES Ex. 367. The perpendiculars drawn from the extremities of one side of a triangle to the median upon that side are equal. Ex. 368. Construct an angle of 75°; of 971°. Ex. 369. Upon a given line find a point such that perpendiculars from it to the sides of an angle shall be equal. Ex. 370. Construct a triangle, given its perimeter and two of its angles. Ex. 371. Construct a parallelogram, given the base, one base angle, and the bisector of the base angle. Ex. 372. Given two lines that would meet if sufficiently prolonged. Construct the bisector of their angle, without prolonging the lines. Ex. 373. Construct a triangle, having given one angle, one adjacent side, and the difference of the other two sides. Case 1: The side opposite the given angle less than the other unknown side. Case 2: The side opposite the given angle greater than the other unknown side. Ex. 374. The difference between two adjacent angles of a parallelogram is 90°; find all the angles. Ex. 375. A straight railway passes 2 miles from a certain town. A place is described as 4 miles from the town and 1 mile from the railway. Represent the town by a point and find by construction how many places answer the description. Ex. 376. Describe a circle through two given points which lie outside a given line, the center of the circle to be in that line. Show when no solution is possible. Ex. 377. Construct a right triangle, given the hypotenuse and the difference of the other two sides. Ex. 378. If two sides of a triangle are unequal, the median through their intersection makes the greater angle with the lesser side. Ex. 379. Two trapezoids are equal if their sides taken in order are equal, each to each. Ex. 380. Construct a right triangle, having given its perimeter and an acute angle. Ex. 381. Draw a line such that its segment intercepted between two given indefinite lines shall be equal and parallel to a given finite line. One angle of a parallelogram is given in position and the point of intersection of the diagonals is given; construct the parallelo Ex. 382. gram. Ex. 383. Construct a triangle, given two sides and the median to the third side. Ex. 384. If from any point within a triangle lines are drawn to the three vertices of the triangle, the sum of these lines is less than the sum of the sides of the triangle, and greater than half their sum. Ex. 385. Repeat the proof of Prop. XIX for two cases at once, using Figs. 1 and 2. B E B E FIG. 2. Ex. 386. If the angle at the vertex of an isosceles triangle is four times each base angle, the perpendicular to the base at one end of the base forms with one side of the triangle, and the prolongation of the other side through the vertex, an equilateral triangle. FIG. 1. Ex. 387. The bisector of the angle C of a triangle ABC meets AB in D, and DE is drawn parallel to AC meeting BC in E and the bisector of the exterior angle at C in F. Prove DE EF. Ex. 388. Define a locus. = Find the locus of the mid-points of all the lines drawn from a given point to a given line not passing through the point. Ex. 389. angle. Ex. 390. side. Construct an isosceles trapezoid, given the bases and one Construct a square, given the sum of a diagonal and one Ex. 391. The difference of the distances from any point in the base prolonged of an isosceles triangle to the equal sides of the triangle is constant. Ex. 392. Find a point X equidistant from two intersecting lines and at a given distance from a given point. Ex. 393. When two lines are met by a transversal, the difference of two corresponding angles is equal to the angle between the two lines. BOOK II THE CIRCLE 276. Def. A circle is a plane closed figure whose boundary is a curve such that all straight lines to it from a fixed point within are equal. 277. Def. The curve which forms the boundary of a circle is called the circumference. 278. Def. The fixed point within is called the center, and a line joining the center to any point on the circumference is called a radius, as QR. Ө R 279. From the above definitions and from the definition of equal figures, § 18, it follows that: (a) All radii of the same circle are equal. (b) All radii of equal circles are equal. (c) All circles having equal radii are equal. 280. Def. Any portion of a circumference is called an arc, as DF, FC, etc. 281. Def. A chord is any straight line having its extremities on the circumference, as DF. 282. Def. A diameter is a chord which passes through the center, as BC. M H T K N 0 B C 283. Since any diameter is twice a radius, it follows that All diameters of a circle are equal. PROPOSITION I. THEOREM 284. Every diameter of a circle bisects the circumference and the circle. A M B N Given circle AMBN with center 0, and AB, any diameter. ARGUMENT 1. Turn figure AMB on AB as an axis until 2. Arc AMB will coincide with arc ANB. 4. Also figure AMB will coincide with figure 5. . figure AMB = figure ANB; i.e. AB bisects circle AMBN. Q.E.D. Ex. 402. A semicircle is described upon each of the diagonals of a rectangle as diameters. Prove the semicircles equal. Ex. 403. Two diameters perpendicular to each other divide a circumference into four equal arcs. Prove by superposition. Ex. 404. Ex. 405. Construct a circle which shall pass through two given points. Construct a circle having a given radius r, and passing through two given points A and B. 287. Def. A sector of a circle is a plane closed figure whose boundary is composed of two radii and their intercepted arc, as sector SOR. 288. Def. A segment of a circle is a plane closed figure whose boundary is composed of an arc and the chord joining its extremities, as segment DCE. 289. Def. A segment which is one half of a circle is called a semicircle, as segment AMB. 290. Def. An arc which is half R M B of a circumference is called a semicircumference, as arc AMB. 291. Def. An arc greater than a semicircumference is called a major arc, as arc DME; an arc less than a semicircumference is called a minor arc, as arc DCE. 292. Def. A central angle, or angle at the center, is an angle whose vertex is at the center of a circle and whose sides are radii. Ex. 406. The line joining the centers of two circles is 6, the radii are 8 and 10, respectively. What are the relative positions of the two circles? Ex. 407. A circle can have only one center. |