SECTION VIII. CONSTRUCTIONS DEPENDING ON ANGLE PROPERTIES. Ex. 1372. Draw a line of 2 ins.; on this line as base draw a triangle with a vertical angle of 40°. (You will find that it is practically impossible to draw the vertical angle directly: first draw the angles at the ends of the base. What is their sum? Notice that many different triangles may be drawn with the given vertical angle.) Ex. 1873. Draw a line of 2 ins.; on this line as base, and on the same side of it, draw a number of triangles (about 10) having a vertical angle of 40°. What is the locus of their vertices? Complete the curve of which this locus is a part. Is it possible for the vertex to coincide with an end of the base, in an extreme case? Does the curve pass through the ends of the base? Ex. 1374. Repeat Ex. 1373 with base 2 in. and vertical angle 140°. Compare this with the locus obtained in Ex. 1373. Ex. 1375. What locus would be obtained if Ex. 1373 were repeated with an angle of 90° ? Ex. 1376. (Tracing paper.) Draw, on tracing paper, two straight lines intersecting at P. On your drawing paper mark two points A, B. Move your tracing paper about so that the one line may always pass through A, and the other through B. Plot the locus of P by pricking through. The foregoing exercises will have prepared the reader for the following statement:— The locus of points (on one side of a given straight line) at which the line subtends a constant angle is an arc of a circle, the given line being the chord of the arc. Ex. 1377. Upon what theorem does the truth of this statement depend? Ex. 1378. What kind of arc is obtained if the angle is (i) acute, (ii) a right angle, (iii) obtuse? Ex. 1379. If the constant angle is 45°, what angle is subtended by the given line at the centre of the circle? Use this suggestion in order to draw the locus of points at which a line of 5 cm. subtends 45°, without actually determining any of the points. Measure the radius. Ex. 1880. By finding the centre directly, construct the locus of points at which a given line subtends an angle of 30°. Prove, and verify by measurement, that in this case the radius of the circle is equal to the given line. Ex. 1881. (Without using protractor.) Construct the locus of points at which a given line subtends a given angle. Ex. 1882. On a chord of 3.5 ins. construct a segment of a circle to contain an angle of 70°. Measure the radius. Ex. 1383. Repeat Ex. 1382 with chord of 7.24 cm. and angle of 110°. Ex. 1384. Repeat Ex. 1382 with chord of 3 in. and angle of 120°. Ex. 1385. Prove that the locus of the mid-points of chords of a circle which are drawn through a fixed point is a circle. Ex. 1386. Of all triangles of given base and vertical angle, the isosceles triangle has greatest area. Ex. 1387. P is a variable point on an arc AB. AP is produced to Q so that PQ=PB. Prove that the locus of Q is a circular arc. To construct a triangle with given base, given altitude, and given vertical angle. Let the base be 7 cm.; the altitude 6.5 cm.; the vertical angle 46°. Draw the given base. Draw the locus of points at which the given base sub tends 46°. Draw the locus of points distant 6.5 cm. from the given base (produced if necessary). The intersections of these loci will be the required positions of the vertex. How many solutions are there to this problem? Ex. 1388. Construct a triangle having (i) base=4 in., altitude=1 in., vertical angle =90°. In each case measure the base angles. Ex. 1389. (Without protractor.) Construct a triangle given the base, vertical angle and altitude. Ex. 1390. Construct a triangle of base 9 cm., vertical angle 47° and median 9 cm. Find the area. (i) base =5 cm., vertical angle=40°, area=16 sq. cm.; (ii) base 4 in., vertical angle=100°, area=2 sq. in. In each case measure the radius of the circumcircle. Ex. 1392. Construct quadrilateral ABCD, given AB = 54 cm., AC 9.5 cm., AD=5'6 cm., LBAD=113°, LBCD=70°. Measure the other diagonal. Ex. 1393. Construct a cyclic quadrilateral ABCD, given AB=1.6 in., BC=3.0 in., CD=4·9 in., <B=125°. Find the radius of the circumcircle. Why are only four measurements given for the construction of this quadrilateral? Ex. 1394. Construct a quadrilateral ABCD, given that AB=6·1 cm., BC=11.4 cm., CA=11·7 cm., AD=5·1 cm., 4BDC=76°. Measure the side CD. Ex. 1395. Construct a parallelogram with base 2.8 in. and height 2 in., the angle between the diagonals being 80°. Measure the other side of the parallelogram. (Try to find the centre of the parallelogram.) To inscribe in a given circle a triangle with given angles. Let the radius of the circle be 2 in. and the angles of the required triangle 40°, 60° and 80°. Analysis Draw a sketch of the required figure; join the vertices of the triangle to the centre of the circle. What are the angles subtended at the centre by the three sides? Knowing these three angles at the centre, it is easy to draw the required figure. Ex. 1396. Draw the figure described above; measure the sides of the triangle. State the construction formally, and give a proof. Ex. 1397. Inscribe in a circle of radius 5 cm. a triangle of angles 30°, 80°, 70°. Measure the sides. Ex. 1398. Inscribe in a circle of radius 3.5 in. a triangle with angles 50°, 40°. Measure the sides. Ex. 1399. Inscribe in a circle of radius 4 cm. an isosceles triangle having each of the angles at the base double the angle at the vertex. Measure the base. Ex. 1400. Inscribe in a circle of radius 2·5 in. a triangle having two of its angles 35° and 40°. Measure the sides. Ex. 1401. (Without protractor.) Inscribe in a circle of radius 6 cm. a triangle equiangular with a given triangle. Check the Ex. 1402. Copy fig. 266 on an enlarged scale; making the radius of the circle 2 in. angles marked, and measure AC. 93 85° D fig. 266. To circumscribe about a given circle a triangle with given angles. Let the radius of the given circle be 24 in.: the angles of the required triangle 45°, 70°, 65°. Analysis Draw a sketch of the required figure (fig. 267). of contact of the sides. If the angles at O can be calculated it will be easy to draw the figure. Now S AMO, ANO are right angles, .. 4S MAN, MON are supplementary. Hence calculate MON, and similarly the other angles at O. Ex. 1403. Draw the figure described above. Measure the longest side of the triangle. State the construction formally, and give a proof. Ex. 1404. Circumscribe about a circle of radius 5 cm. a triangle of angles 30°, 60°, 90°. Measure the longest side. Ex. 1405. Circumscribe about a circle of radius 3 in. an isosceles right-angled triangle. Measure the longest side. Ex. 1406. Circumscribe about a circle of radius 4 cm. a parallelogram having an angle of 70°. Measure the sides of the parallelogram, and prove that it is a rhombus. Ex. 1407. (Without protractor.) Circumscribe about a circle of radius 2.6 in. a triangle equiangular to a given triangle. Ex. 1408. (Without protractor.) Circumscribe about a circle of radius 5 cm. a triangle having its sides parallel to three given straight lines. SECTION IX. "ALTERNATE SEGMENT." THEOREM 14. If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments. Data AB touches OCDE in c; the chord CD is drawn through C, meeting again in D. (1) Construction Through C draw CE to AB, meeting in E. Join CE, DE. Proof Since CE is drawn to tangent AB, at its point of contact C, .. CE passes through centre of O, and is a diameter, .. LCDE is a rt. ▲, .. in ACDE, CED+ DCE = 1 rt. 4. .. 4 BCD + DCE = CED + 4 DCE, ... 4 BCDL CED. III. 6, Cor. III. 10. I. 8. Constr. |