Ex. 710. By means of § 448, construct a line equal to √2 inches. Ex. 711. If a side of a square is 6 inches, find its diagonal. Ex. 712. The hypotenuse of a right triangle is 15 and one arm is 9. Find the other arm and the segments of the hypotenuse made by the perpendicular from the vertex of the right angle. Ex. 713. Find the altitude of an equilateral triangle whose side is 6 inches. Ex. 714. 8 inches. Ex. 715. of 1 to √2. Find a side of an equilateral triangle whose altitude is Divide a line into segments which shall be in the ratio Ex. 716. The radius of a circle is 10 inches. Find the length of a chord 6 inches from the center; 4 inches from the center. Ex. 717. The radius of a circle is 20 inches. How far from the center is a chord whose length is 32 inches? whose length is 28 inches ? Ex. 718. In a circle a chord 24 inches long is 5 inches from the center. How far from the center is a chord whose length is 12 inches? 450. Def. The projection of a point upon a line is the foot of the perpendicular from the point to the line. 451. Def. The projection of a line segment upon a line is the segment of the second line included between the projections of the extremities of the first line upon the second. Thus, C is the projection of 4 upon MN, D is the projection of B upon MN, and CD is the projection of AB upon MN. Ex. 719. In the figures above, under what condition will the projection of AB on MN be a maximum ? a minimum ? CD ever be equal to AB? greater than AB? be a point? Will the projection Will the projection ever Ex. 720. In a right isosceles triangle the hypotenuse of which is 10 inches, find the length of the projection of either arm upon the hypotenuse. Ex. 721. Find the projection of one side of an equilateral triangle upon another if each side is 6 inches. Ex. 722. Draw the projections of the shortest side of a triangle upon each of the other sides: (1) in an acute triangle; (2) in a right triangle; (3) in an obtuse triangle. Draw the projections of the longest side in each case. Ex. 723. Two sides of a triangle are 8 and 12 inches and their included angle is 60°. Find the projection of the shorter upon the longer. Ex. 724. In Ex. 723, find the projection of the shorter side upon the longer if the included angle is 30°; 45°. Ex. 725. Parallel lines that have equal projections on the same line are equal. PROPOSITION XXVIII. PROBLEM 452. In any triangle to find the value of the square of the side opposite an acute angle in terms of the other two sides and of the projection of either of these sides upon the other. A A Given ▲ BAX, with X acute; and p, the projection of b upon a. To find the value of a2 in terms of a, b, and p. 3. § 54, 11. 4. § 54, 13. 5. § 309. Q.E.F. 3. And DB=a-p (Fig. 1) or p—a (Fig. 2). .. x2= b2 - p2 + a2 - 2 ap + p2; i.e. x2= a2 + b2 — 2 ap. 453. Question. Why is it not necessary to include here the figure and discussion for a right triangle? 454. Prop. XXVIII may be stated in the form of a theorem as follows: In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of these sides and the projection of the other side upon it. Ex. 726. If the sides of a triangle are 7, 8, and 10, is the angle opposite 10 obtuse, right, or acute ? Why? Ex. 727. Apply the statement of Prop. XXVIII to the square of an arm of a right triangle. Ex. 728. Find x (in the figure for Prop. XXVIII) in terms of a and b and the projection of a upon b. Ex. 729. If the sides of a triangle are 13, 14, and 15, find the projection of the first side upon the second. Ex. 730. If two sides of a triangle are 4 and 12 and the projection of the first side upon the second is 2, find the third side of the triangle. PROPOSITION XXIX. THEOREM 455. In any obtuse triangle, the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides increased by twice the product of one of these sides and the projection of the other side upon it. Given ▲ BAX with X obtuse, and p, the projection of b upon a. To prove x2= a2 + b2 + 2 ap. 456. From Props. XXVIII and XXIX, we may derive the following formulas for computing the projection of one side of a triangle upon another; thus if a, b, and c represent the sides of a triangle: It is seen that the second members of these two equations are identical and that the first members differ only in sign. Hence, formula (1) may always be used for computing the length of a projection. It need only be remembered that if p is positive in any calculation, it indicates that the angle opposite c is acute; while if p is negative, the angle opposite c is obtuse. It can likewise be shown (see Prop. XXVII) that if p = 0, the angle opposite c is a right angle. Ex. 731. Write the formula for the projection of a upon b. Ex. 732. In triangle ABC, a = 15, b = 20, c = 25; find the projection of b upon c. Is angle A acute, right, or obtuse? Ex. 733. In the triangle of Ex. 732, find the projection of a upon b. Is angle Cacute, right, or obtuse? Ex. 734. The sides of a triangle are 8, 14, and 20. Is the angle opposite the side 20 acute, right, or obtuse? Ex. 735. If two sides of a triangle are 10 and 12, and their included angle is 120°, what is the value of the third side? Ex. 736. If two sides of a triangle are 12 and 16, and their included angle is 45°, find the third side. Ex. 737. If in triangle ABC, angle C = 120°, prove that AB2 = BC2 + AC2 + AC· BC. Ex. 738. If a line is drawn from the vertex C of an isosceles triangle ABC, meeting base AB prolonged at D, prove that 457. In any triangle, the sum of the squares of any two sides is equal to twice the square of half the third side increased by twice the square of the median upon that side. A Given ▲ ABC with ma, the median to side a. 2 B 2(1) * a +2 m2. ARGUMENT REASONS 1. Suppose b>c; then ADC is obtuse 1. § 173. and BDA is acute. 2. Let p be the projection of m, upon a. 3. Then from ▲ ADC, 2. § 451. 3. § 455. |