Let fall a perpendicular from the angle opposite the greater side, dividing the given triangle into two rightangled triangles: then find the difference of the segments of the base by Theorem IIJ. Half this difference being added to half the base, gives the greater segment; and, being subtracted from half the base, gives the less segment. Then, since the greater segment belongs to the right-angled triangle having the greater hypothenuse, we have two sides and the right angle of each of two right-angled triangles, to find the acute angles. In the triangle DAC, to find the angle DAC. In the triangle BAD, to find the angle BAD. = Hence, 90° - DAC-90° - 51° 34′.40" 38° 25′ 20′′ = 0, = B, and, 90° · BAD = 90° - 32° 18′ 35′′ 57° 41' 25": and, BAD +DAC=51° 34′ 40′′ +32° 18′ 35′′ = 83° 53′ 2. In a triangle, of which the sides are 4, 5, and 6, what are the angles? Ans. 41° 24′ 35′′; 55° 46′ 16′′; and 82° 49′ 09". SOLUTION OF RIGHT-ANGLED TRIANGLES. 34. The unknown parts of a right-angled triangle may be found by either of the four last cases; or, if two of the sides are given, by means of the property that the square of the hypothenuse is equivalent to the sum of the squares of the two other sides. Or the parts may be found by Theorems IV. and V. EXAMPLES. 1. In a right-angled triangle BAC, there are given the hypothenuse BC= 250, and the base AC= 240 required the other parts. Ans. B = 73° 44′ 23′′; C=16° 15′ 37′′; AB=70.0003. B A 2. In a right-angled triangle BAC, there are given AC=384, and B = 53° 08′: required the remaining parts. Ans. AB 287.96; BC=479.979; C=36° 52'. = APPLICATION TO HEIGHTS AND DISTANCES. 1. A HORIZONTAL PLANE is one which is parallel to the water level. 2. A plane which is perpendicular to a horizontal plane, is called a vertical plane. 3. All lines parallel to the water level, are called hori zontal lines. 4. All lines which are perpendicular to a horizontal plane, are called vertical lines; and all lines which are inclined to it, are called oblique lines. 5. A HORIZONTAL ANGLE is one whose sides are hori zontal. 6. A VERTICAL ANGLE is one, the plane of whose sides is vertical. 7. An angle of elevation, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side. 8. An angle of depression, is a vertical angle having one of its sides horizontal, and the inclined side under the horizontal side. I. To determine the horizontal distance to a point which is inaccessible by reason of an intervening river. 35. Let C be the point. Measure along the bank of the river a hori zontal base line AB, and select the B Let us suppose that we have found AB 600 yards, CAB = 57° 35', and CBA = 64° 51'. = The angle C 180° — (A + B)= 57° 34'. To find the distance BC. II. To determine the altitude of an inaccessible object above given horizontal plane. FIRST METHOD. 36. Suppose D to be the inaccessible object, and BC the horizontal plane from which the altitude is to be estimated: then, if we suppose DC to be a vertical line, it will represent the required altitude. B Measure any horizontal base line, as BA; and at the extremities B and A, measure the horizontal angles CBA and CAB. Measure also the angle of elevation DBC. Then in the triangle CBA there will be known, two angles and the side AB; the side BO can therefore be determined. Having found BC, we shall have, in the right-angled triangle DBC, the base BC and the angle at the base, to find the perpendicular DC, which measures the altitude of the point D above the horizontal plane BC. Let us suppose that we have found BA=780 yards, the horizontal angle CBA = 41° 24'; the horizontal angle CAB=96° 28′, and the angle of eleva tion DBC=10°43'. In the triangle BCA, to find the horizontal distance BC. The angle BCA = 180° — (41° 24′ +96° 28′) = 42° 08′ = C. In the right-angled triangle DBC, to find DC. REMARK I. It night, at first, appear, that the solution which we have given, requires that the points B and A should be in the same horizontal plane; but it is entirely independent of such a supposition. For, the horizontal distance, which is represented by BA, is the same, whether the station A is on the same level with B, above it, or below it. The horizontal angles CAB and CBA are also the same, so long as the point C is in the vertical line DC. Therefore, if the horizontal line through A should cut the vertical line DC, at any point, as E, above or below C, AB would still be the hori zontal distance between B and A, and AE, which is equal to AC, would be the horizontal distance between A and C. If at A, we measure the angle of elevation of the point D, we shall know in the right-angled triangle DAE, the base AE, and the angle at the base; from which the per pendicular DE can be determined. 37. Let us suppose that we had measured the angle of elevation DAE, and found it equal to 20° 15'. First: In the triangle BAC, to find AC or its equal AE. In the right-angled triangle DAE, to find DE. |