Express each of the following theorems (Ex. 1090—1093) as an algebraical identity; prove the identity. Ex. 1090. If there are two straight lines, one of which is divided into any number of parts (x, y, z say) while the other is of length a, then the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the undivided straight line and the several parts of the divided line. (Draw a figure.) Ex. 1091. If a straight line is divided into any two parts (x and y), the square on the whole line is equal to the sum of the rectangles contained by the whole line and each of the parts. (Draw a figure.) Ex. 1092. If a straight line is divided into any two parts, the rectangle contained by the whole line and one of the parts is equal to the square on that part together with the rectangle contained by the two parts. (Draw a figure.) Ex. 1093. If a straight line is divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the two parts. (Draw a figure.) Ex. 1094. What algebraical identity is suggested A P by fig. 202? (Take AO=OB=a, OP=b.) Ex. 1095. Express and prove algebraically :—If a straight line is divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, is equal to the square on the straight line which is made up of the whole line and the first part. fig. 202. Ex. 1096. Prove that the square on the difference of the sides of a right-angled triangle, together with twice the rectangle contained by the sides, is equal to the square on the hypotenuse. (Use Algebra.) Ex. 1097. If a straight line AB (length 2x) is bisected at O and also divided unequally at a point P (distant y from O), what are the lengths of the two unequal parts AP, PB? Prove algebraically that the rectangle contained by the unequal parts, together with the square on the line between the points of section (OP), is equal to the square on half the original line. Ex. 1098. Show that in the above exercise AO is half the sum of AP, PB; and that OP is half the difference of AP, PB. (Most easily proved by Algebra.) Ex. 1099. If a straight line AB (length 2x) is bisected at O, and produced to any point P(OP=y) the rectangle contained by the whole line thus produced and the part of it produced, together with the square on half the original line, is equal to the square on the straight line made up of the half and the part produced. Ex. 1100. If a straight line is divided into any two parts, the sum of the squares on the whole line and on one of the parts is equal to twice the rectangle contained by the whole and that part, together with the square on the other part. (Draw figure.) Ex. 1101. If a straight line AB is bisected at O and also divided unequally at a point P (as in Ex. 1097), the sum of the squares on the two unequal parts is twice the sum of the squares on half the line and on the line between the points of section (OP). Ex. 1102. If a straight line is bisected and produced to any point (as in Ex. 1099), the sum of the squares on the whole line thus produced and on the part produced, is twice the sum of the squares on half the original line, and on the line made up of the half and the part produced. Ex. 1103. Four points A, B, C, D are taken in order on a straight line; prove algebraically that AB. CD + BC. AD = AC. BD. (Take AB = x, BC=y, CD=z.) Verify numerically. Ex. 1104. If a straight line is bisected and also divided unequally (as in Ex. 1097) the squares on the two unequal parts are together equal to twice the rectangle contained by these parts together with four times the square on the line between the points of section. 14 G. S. PROJECTIONS. DEF. If from the extremities of a line AB perpendiculars AM, BN are drawn to a straight line CD, then MN is called the projection of AB upon CD (figs. 203, 204). Ex. 1105. In fig. 189 name the projection of AB upon DE; of AE upon BC; of AC upon AL. NC. Ex. 1106. In fig. 208 name the projection of AC upon BN; of BC upon Ex. 1107. (On squared paper.) What is the length of the projection (i) upon the axis of x, (ii) upon the axis of y, of the straight lines whose extremities are the points (a) (2, 3) and (6, 6). (b) (2, 4) and (6, 7). Ex. 1108. (c) (0, 0) and (4, 3). (d) (-1, -3) and (3, 0). (g) (0, -2) and (0, 2). Prove that the projections of equal and parallel straight lines are equal. (See fig. 205.) Ex. 1109. O is the mid-point of AB; the pro jections of A, B, O upon any line are P, Q, T. Prove that PT=QT. Ex. 1110. Measure the projection of a line of fig. 205. length 10 cm. when it makes with the line upon which it is projected the following angles : 15°, 30°, 45°, 60°, 75°, 90°. Ex. 1111. In what case is the projection of a line equal to the line itself? Ex. 1112. In what case is the projection of a line zero? Ex. 1113. Prove that, if the slope of a line is 60°, its projection is half the line. [Consider an equilateral triangle.] Ex. 1114. A pedestrian first ascends at an angle of 12° for 2000 yards and then descends at an angle of 9° for 1000 yards. How much higher is he than when he started? What horizontal distance has he travelled (i.e. what is the projection of his journey on the horizontal)? Ex. 1115. angles are x, y. The projections of a line of length l upon two lines at right Ex. 1116. How does the projection of a line of given length alter as the slope of the line becomes more and more steep? NOTE. It may be necessary to produce the line upon which we project, e.g. if required to project AB upon MN in fig. 206, we must produce CD. D M fig. 206. EXTENSION OF PYTHAGORAS' THEOREM. BAC, BAC1, BAC2 (fig. 207) are triangles respectively rightangled, acute-angled, and obtuse-angled A fig. 207. and .. BC12=C1A2+ AB2 some area, BC2=C2A2 + AB2 + some area. The precise value of the quantity referred to as is given in the two following theorems. THEOREM 8. In an obtuse-angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle plus twice the rectangle contained by one of those sides and the projection on it of the other. Data fig. 208. The AABC has 4 BAC obtuse. CN is the perpendicular from C upon BA (produced), |