PROPOSITION XXXIV. PROBLEM 325. To construct the mean proportional between two given lines. Required the mean proportional between m and n. Construction I. Draw AB=m, and produce AB to C so that BC=n. On AC as a diameter, describe a semicircle. At B, erect a perpendicular upon AC, meeting the circle in D. BD is the required mean proportional. [The proof is left to the student.] Construction II. Draw AB = n. On AB as a diameter, describe a semicircle. On AB lay off AC: = m. Draw CE LAB. Join A and E. AE is the required mean proportional. [The proof is left to the student.] m A Ex. 917. If a and b are given lines, construct √ab. Ex. 918. Construct √6 ab if a and b are two given lines. B PROPOSITION XXXV. THEOREM 326. The sum of the squares of the arms of a right triangle is equal to the square of the hypotenuse. a Given ABC a rt. ▲, having its rt. ≤ at C. To prove Proof. a2 + b2 = c2. Draw CDL AB, and denote AD by p, and DB by q. 327. COR. The square of either arm of a right triangle is equal to the square of the hypotenuse, diminished by the square of the other arm. Ex. 922. Find the hypotenuse of a right triangle whose arms are respectively (a) 1 ft. and 5 in. (b) m and n. Ex. 923. The hypotenuse of a right triangle is 25, one arm equals 20; find the other arm. Ex. 924. Find the altitude of an equilateral triangle whose side is equal to 8 in. Ex. 925. Find the altitude of an equilateral triangle whose side is equal to a. Ex. 926. Find the altitude of an isosceles triangle whose base equals 8 and whose arm equals 5. Ex. 927. If the hypotenuse of an isosceles right triangle equals 8 in., what is the length of an arm? and Ex. 928. The radii of 2 circles are respectively 6 in. and 21 in., the distance between their centers is 25 in.; find the length of the common external tangent. Ex. 929. Find the side of an equilateral triangle whose altitude equals 10. Ex. 930. The squares of the two arms of a right triangle have the same ratio as the adjacent segments of the hypotenuse. Ex. 931. If AD is an altitude of a triangle ABC, Ex. 932. If the diagonals of a quadrilateral are perpendicular to each other, then the sum of the squares of two opposite sides is equal to the sum of the squares of the other two. Ex. 933. If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is a right triangle. HINT. Draw a rt. A whose arms are respectively equal to the arms of the given ▲, and prove the equality of the two A. *Ex. 934. If the square of one side of a triangle is greater than the sum of the squares of the other two, the triangle is obtuse. HINT. Compare the ▲ with a rt. ▲ which has the same arms as the given one. 328. DEF. The projection of a point upon a line is the foot of the perpendicular from the point to the line. 329. DEF. The projection of one line upon another is the segment between the projections of the extremities of the first line upon the second. Thus, if AA' 1 XY and BB' 1 XY, A' is the projection of A upon XY, and A'B' is the projection of AB upon XY. Ex. 935. In acute triangle ABC, draw the projection of AB upon AC, of AB upon BC, of AC upon AB. Ex. 936. If AB|| XY, prove that the projection of AB upon XY equals AB. Ex. 937. If the side of an equilateral triangle equals 10 in., what is the length of the projection of one side upon another? A A 60° X A B' Ex. 938 B' Ex. 939 Ex. 938. If the lines AB and XY include an angle of 60°, the projection of AB upon XY equals one half AB (167). Ex. 939. Prove that the projection A'B' of AB upon XY equals one half AB, if the prolongation of BA forms an angle of 60° with XY. ДА 30° Ex. 940 A 45° B' Ex. 941 Ex. 940. If the lines AB and XY include an angle of 30°, and AB= m, prove that the projection of AB upon XY equals √3. m an angle of 45°, and Ex. 941. If the lines AB and XY include HINT. If AB' = x, then BB' = x. .. x2 + x2 = m2. m √2. Ex. 942. Prove the last two exercises, if the prolongation of AB forms an angle of 30°; an angle of 45°. (Diagram similar to Ex. 939.) Ex. 943. Find the projection of AB upon XY if AB = m, and the angle included by AB and XY equals 120°. Ex. 944. Find the projection of AB upon XY if_AB = m, and the angle included by AB and XY equals 135°. Ex. 945. Find the projection of AB upon XY if AB = m, and the angle included by AB and XY equals 150°. = Ex. 946. If in triangle ABC, AB = 8, AC the projection of AB upon AC, of BC upon AC. 10, and A = 60°, find Ex. 947. If in triangle ABC, AB=10, AC 12, and A = 45°, find the projection of AB upon AC. Ex. 948. In the same figure find the projection of BC upon AC. Ex. 949. In triangle ABC, AC = 24, BC = 10, and ▲ C = 90°. Find the projection of AC upon AB. [See practical applications, p. 299.] 330. NOTE. A abc denotes a triangle whose sides are a, b, and c. p denotes the projection of b upon c, and q the projection of a upon c. The other notations used in the following propositions are in accordance with § 244. 8, find q. a Ex. 950. In ▲ abc, if b = 4, p = 2, find ▲ A and he. Ex. 951. Ex. 955. In A abc, if a = = 20, b = 37, q = = 16, find p. 15, p = 9, and c = 25, find a. Ex. 957. In ▲ abc, express a in terms of b, c, and p. |