Ex. 763. HINT. In P'A AD. Ex. 764. and if s is the HINT. Ex. 765. mean ratio. Ex. 766. mean ratio. Divide a line AB externally in extreme and mean ratio. the figure for Prop. XXXIII prolong BA to P', making Then prove AB : P'A = P'A : P'B. If the line is divided internally in extreme and mean ratio, A line 10 inches long is divided internally in extreme and A line 8 inches long is divided externally in extreme and E Ex. 768. A little boy wished to obtain the height of a tree in his yard. He set up a vertical pole 6 feet high and watched until the shadow of the pole measured exactly 6 feet. He then measured quickly the length of the tree's shadow and called this the height of the tree. Was his answer correct? Draw figures and explain. Use this method for measuring the height of your school building and flag pole. E W Ex. 769. If light from a tree, as AB, is allowed to pass through a small aperture O, in a window shutter W, and strike a white screen or wall, an inverted image of the tree, as CD, is formed on the screen. If the distance OE30 feet, OF=8 feet, and the length of the tree AB=35 feet, find the length of the image CD. Under what condition will the length of the image equal the length of the tree? This exercise illustrates the principle of the photographer's camera. B Ex. 770. By means of Prop. XXXII construct a mean proportional between two given lines. Ex. 771. In a certain circle a chord 5√5 inches from the center is 20 inches in length. Find the length of a chord 9 inches from the center. Ex. 772. Compute the length of: (1) the common external tangent, (2) the common internal tangent, to two circles whose radii are 8 and 6, respectively, and the distance between whose centers is 20. Ex. 773. If the hypotenuse of an isosceles right triangle is 16 inches, what is the length of each arm? Ex. 774. If from a point a tangent and a secant are drawn and the segments of the secant are 4 and 12, how long is the tangent ? Ex. 775. Given the equation m + n = 2 m solve for x. Ex. 776. Find a mean proportional between a2+2ab+b2 and a2 - 2 ab + b2. Ex. 777. The mean proportional between two unequal lines is less than half their sum. B Ex. 778. The diagonals of a trapezoid divide each other into segments which are proportional. Ex. 779. ABC is an isosceles triangle inscribed in a circle. Chord BD is drawn from the vertex B, cutting the base in any point, as E. Prove BD: AB=AB: BE. E D Ex. 780. In a triangle ABC the side AB is 305 feet. If a line parallel to BC divides AC in the ratio of 2 to 3, what are the lengths of the segments into which it divides AB? Ex. 781. Construct, in one figure, four lines whose lengths shall be that of a given unit multiplied by √2, √3, 2, √б, respectively. Ex. 782. Two sides of a triangle are 12 and 18 inches, and the perpendicular upon the first from the opposite vertex is 9 inches. What is the length of the altitude upon the second side? : (a + 1/2 a) ; (b + 1/2v) = (c + 2 c) : (a + 2a). m m Also translate this fact into a verbal statement. Ex. 784. If a constant is added to or subtracted from each term of a proportion, will the resulting numbers be in proportion? Give proof. Ex. 785. If r: st: q, is 3 r + :s=7t: 2q? Prove your answer. 2 Ex. 786. One segment of a chord drawn through a point 7 units from the center of a circle is 4 units long. If the diameter of the circle is 15 units, what is the length of the other segment? Ex. 787. The non-parallel sides of a trapezoid and the line joining the mid-points of the parallel sides, if prolonged, are concurrent. Ex. 788. Construct a circle which shall pass through two given points and be tangent to a given straight line. Ex. 789. The sides of a triangle are 10, 12, 15. Compute the lengths of the two segments into which the least side is divided by the bisector of the opposite angle. Ex. 790. AB is a chord of a circle, and CE is any chord drawn through the middle point C of arc AB, cutting chord AB at D. Prove CE: CA = CA: CD. Ex. 791. Construct a right triangle, given its perimeter and an acute angle. C A B D E Ex. 792. The base of an isosceles triangle is a, and the perpendicular let fall from an extremity of the base to the opposite side is b. Find the lengths of the equal sides. Ex. 793. AD and BE are two altitudes of triangle CAB. Prove that AD: BE CA: BC. = Ex. 794. If two circles touch each other, their common external tangent is a mean proportional between their diameters. Ex. 795. If two circles are tangent internally, all chords of the greater circle drawn from the point of contact are divided proportionally by the circumference of the smaller circle. Ex. 796. If three circles intersect each other, their common chords pass through a common point. Ex. 797. The square of the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments of the third side made by the bisector. Given ▲ ABC with t, the bisector of B, dividing side b into the two segments s and r. B a A Dlt 8 0 m E Ex. 798. In any triangle the product of two sides is equal to the product of the altitude upon the third side and the diameter of the circumscribed circle. HINT. Prove ▲ ABD~ EBC. Then prove B a A E BOOK IV AREAS OF POLYGONS 466. A surface may be measured by finding how many times it contains a unit of surface. The unit of surface most frequently chosen is a square whose side is of unit length. If the unit length is an inch, the unit of surface is a square whose side is an inch. Such a unit is called a square inch. If the unit length is a foot, the unit of surface is a square whose side is a foot, and the unit is called a square foot. 467. Def. The result of the measurement is a number, which is called the measure-number, or numerical measure, or area of the surface. 468. Thus, if the square U is contained in the rectangle ABCD (Fig. 1) 15 times, then the measure-number or area of rectangle ABCD, in terms of U, is 15. If the given square is not contained in the given rectangle an integral number of times without a remainder (see Fig. 2), then by taking a square which is an aliquot part of U, as one fourth of U, and applying FIG. 3. Rectangle EG U+= 11} U+. it as a measure to the rectangle (see Fig. 3) a number will be obtained which, divided by four,* will give another (and G A FIG. 4. Rectangle AG = 90 U+ =11} U+. usually closer) approximate area of the given rectangle. By proceeding in this way (see Fig. 4), closer and closer approximations of the true area may be obtained. *It takes four of the small squares to make the unit itself. |