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92. If the opposite angles of a quadrilateral be equal, the opposite sides are equal.
93. The straight lines which join the middle points of the opposite sides of a quadrilateral, and the straight line which joins the middle points of the diagonals, are all concurrent and bisect each other.
94. Prove that the angle between the bisectors of two adjacent angles of a quadrilateral is half the sum of the two remaining angles of the quadrilateral.
95. A quadrilateral has two sides parallel and two sides equal but not parallel, show that the diagonals of the quadrilateral are equal.
96. If two opposite sides of a parallelogram be bisected and two straight lines be drawn from the points of bisection to two opposite angles, the two lines trisect the diagonal which passes through the other two angular points of the parallelogram.
97. If two qualrilaterals have three sides of the one equal to three sides of the other, each to each, and have also the angles contained by those sides equal, each to each, then the quadrilaterals shall be equal in all respects.
VIII.-On Areas and Squares. 98. Of all the parallelograms which can be formed with diameters of given lengths, the rhombus is the greatest.
99. Two rectangles are equal in area if two adjacent sides of the one are equal to two adjacent sides of the other.
100. The area of a triangle (or parallelogram) is equal to the sum or difference of the areas of two triangles (or parallelograms) on the same base (or equal bases) if the altitude of the former is equal to the sum or difference of the altitudes of the latter.
101. ABCD is a parallelogram and P is any point; show that the sum or difference of the areas of the triangles PAB, PAD is equal to the area of the triangle PAC. Distinguish the cases for the sum and difference.
102. ABC is a given triangle and P is a point in its base; if two lines be drawn through P parallel to the sides of the triangle, show that the parallelogram so formed is greatest when P is taken at the middle of the base.
103. The area of a qnadrilateral is equal to the area of a triangle, two of whose sides are equal to the diagonals of the quadrilateral, the angle included by these sides being also equal to the angle between the diagonals.
104. If a square be inscribed in a triangle, twice the area of the triangle will be equal to the rectangle contained by a side of the square and a line equal to the sum of the bil.sc and altitude of the triangle.
105. Through one angle of a triangle draw a straight line to the opposite side cutting off from the triangle any given area.
106. Bisect a quadrilateral by a straight line drawn (i) through one of its angles, or (ii) through a given point in one of its sides.
107. Trisect a parallelogram by straight lines drawn (i) through one of its angles, or (ii) through a given point in one of its sides.
108. If a straight line be divided into any two parts, the square on the whole line is greater than the sum of the squares on the two parts.
109. Divide a given straight line into two parts so that the square on one part may be double the square on the other part.
110. Divide a straight line into two parts, the difference of whose squares shall be equal to a given square.
111. In a right-angled triangle the equilateral triangle described on the hypotenuse is equal to the sum of the equilateral triangles described on the two sides.
112. If any parallelograms ABDE, ACFG be described externally on the sides AB, AC of any triangle ABC, and if DE, FG, or these lines produced, meet in H; then the parallelogram BKLC, described on BC and having its sides BK, CL equal and parallel to HA, will be equal to the sum of the parallelograms AD and AF.
PRACTICAL EXERCISES ON BOOK I.
Note on Euclid's Constructions.
In practical work Propositions 9, 10, 11, 22, 31 of Book I. should be replaced by the methods given in the Preliminary Course in Problems V., IV., II., XIV. respectively.
Prop. 23 is usually modified as follows :—With centre D describe an arc cutting DC in H and DE in K; with centre A and the same radius describe an arc cutting AB in L; from the latter arc cut off a portion LM equal to the arc HK, measuring by the compasses ; join AM.
Problems on Propositions 1-15.
1. Draw the figure of I. 15, making _DEB = 40°. Measure 28 DEA, AEC, CEB. Bisect each of the four angles at E, and verify that the bisectors form two perpendicular straight lines. [Cf. Props. 13, 15, and Ex. 1, p. 40.]
2. Draw a line AB of length 1.5". Draw the locus of all points distant 1" from A; also the locus of all points distant 1.5" from B. Find two points, each of which lies 1" from A and 1.5" from B.
3. Draw a line AB of length 30 mm. Find two points, each of which lies 20 mm. from A and from B; also two points, each of which lies 35 mm. from A and from B.
4. Draw a line AB of length 30 mm. Draw the locus of all points which are equidistant from A and B. 5. Construct a A ABC such that AB
BC 4 cm., CA
= 4 cm. Draw the locus of all points which are equidistant from A and B; also the locus of all points which are equidistant from B and C. Find a point which is equidistant from A, B, and C, and measure its distance from any one of these points.
6. Construct a A ABC such that AB = 2", BC = 2.5", CA = 1.5". Find a point on AB which is equidistant from B and C, and measure its distance from either. EUC, 113
Problems on Propositions 1-34.
7. Verify Euclid I. 24 in the case of two triangles each of which has two sides of 30 and 50 mm. length respectively, the included angle in the first being 90° and in the second 135o.
8. Draw a triangle ABC such that AB = 2", BC = 2.5", CA = 3". Bisect AB and AC at M and N respectively. Draw AD I BC and meeting BC in D; also draw ME and NF 1 BC. Now cut the triangle out with a penknife. Fold each of the corners A,B,C to the point D. (The creases of the paper will be MN, ME, and NF). Hence show that the angles A, B, C are together equal to two right angles (as they are equal to the three angles MDN, MDE, NDF respectively). This construction will hold for any triangle, but if the triangle is obtuse BC must be the longest side.
9. Draw the locus of all points which are •7" distant from a given line AB (of indefinite length). [The locus consists of two lines || to AB, one on each side.]
10. Draw two lines AOB, COD intersecting at 0, such that LAOC - 70°. Draw the locus of all points which are •6" from AOB, and the locus of all points which are •9" from COD. Mark 4 points, each of which is .6" from AOB and .9" from COD.
11. Draw two lines AOB, COD as in Question 10. Draw the locus of all points which are equidistant from these lines. (The locus consists of the lines bisecting each of the angles formed at 0.) 12. Draw a triangle ABC, such that AB
BC = 4 cm., CA 5 cm. Find two points, one in CB and one in CB produced, which are equidistant from AB and AC. Measure the distance of each point from AC.
13. Draw a triangle ABC as in question 12. Find a point within the triangle equidistant from all three sides. Measure its distance from either side.
14. Draw an acute-angled triangle on a base 2", of altitude 1", having one side = 1.3". Measure the other side.
15. Draw an obtuse-angled triangle, having the measurements given in question 14. Measure the other side.
Problems on Propositions 1-48.
16. Show by measurement that if the sides of a right-angled triangle are 3 and 4 cm, respectively, its hypotenuse is 5 cm. [Cf. Prop. 47.]
17. Show by measurement that the triangle whose sides are •5", 1.2", 1•3" respectively is right-angled. [Cf. Prop. 48.]
18. Show by measurement that the triangle whose sides are •5", 1.2", 1•4" respectively is obtuse-angled. [Cf. question 17, and Prop. 25.]
19. Show by construction that the area of a rectangle whose adjacent sides are 3 and 5 cm. is 15 square centimetres.
20. Show by construction that the area of a rectangle whose adjacent sides are " and }" is square inch.
21. Show by construction (using question 20) that the area of a rectangle whose sides are 21" and 2}" is thirty-five sixths of a square inch.
22. Assuming that the area of a rectangle is in all cases given by the product of two adjacent sides (compare questions 19, 20, 21), prove that
(i) area of a parallelogram = base x altitude. [Cf. Prop. 35.] (ii) area of a triangle = 1 base x altitude. [Cf. Prop. 41.] (iii) area of a right-angled triangle = product of the sides containing
the right angle. [Cf. Prop. 34.] 23. Construct again the first four triangles given on page ix and calculate their areas.
24. Construct again the first four quadrilaterals given on page xx; reduce each to a triangle by the method of page 83, and thence calculate their
an area =
25. Draw a line 7 cm. long. If this line is the base of a triangle whose area is 14 sq. cm., draw the locus of the vertex.
26. Construct on isosceles triangle having a base of 3.6" and an area : 4:32 sq. inches. Measure the sides. [First calculate its altitude.] 27. Construct a parallelogram having a base = 2.2", an angle = 70°, and
3.3 sq. inches. Measure the side. 28. Construct a triangle having an area of 15 sq. cm., an angle = 50°, and one side adjacent to this angle
Measure the side opposite to this angle.
29. Construct a triangle in which two sides measure 4 and 4.5 cm. respectively, and the area is 7 sq. cm. Show that from these data two different triangles can be drawn, and measure the third side in each case.
30. Construct a rectangle of sides 20 and 30 mm. Construct another rectangle of equal area having one side = 16 mm. [Use Prop. 44.] Measure the other side and verify by calculation.
31. If in the figure of Prop. 44 (p. 86), ZGBE 90°, BG = p inches, BE
q inches, BA = q inches, prove that BM = pxq:r inches. 32. Use the result of question 31 to evaluate the following results by geometrical construction. [Note that we need not reproduce the whole figure of Prop. 44. To evaluate pxq=r;—make HG = r, ZHGB = 90°; draw BE = q and || to HG; draw EK || to GB meeting HB produced at K; measure EK.] (i) 16 x 30:20 [work in mm.
2.]; (ii) 1.5x3:2-4 [work in inches]; (iii) 2.1 x 3.2 [work in cm. and make r = 1]; (iv) 2:31:3 [work in inches and make p = 1]; (v) 1:5 x 4•1•3.2; (vi) 2.3 x 3•4; (vii) 4:5:1.8; (viii) 7.3-4.2.
GB = P,