To prove Нур. d is a diameter of the circle circumscribed about A abc, and h is the altitude upon c. ab = hd. [The proof is left to the student.] 331. COR. The diameter of the circumscribed circle of any triangle is equal to the product of two sides divided by the altitude upon the third side. ab abc (a=* or d = Ex. 668. A ABC is inscribed in a circle of radius = 5 inches. Find the altitude to BC' if AB = 4, and AC = 5. Ex. 669. The three sides of a triangle are 4, 13, and 15, respectively. Find the radius of the circumscribed circle. Ex. 670. In A abc, a = 20, b = 15, and the projection of b upon c equals 9. Find the radius of the circumscribed circle. Ex. 671. In Aabc, a = 9 and b = 12. Find c if the diameter of the circumscribed circle equals 15. PROBLEMS OF COMPUTATION Ex. 672. A straight line AB = 4, is divided externally in the ratio 5 : 4. Find the segments. Ex. 673. The shadow of a church steeple upon level ground is 60 ft., while a pole 10 ft. high casts a shadow 3 ft. long. How high is the steeple ? Ex. 674. The arms of a right triangle are 8 and 15 respectively. Compute the hypotenuse and the altitude upon the hypotenuse. Ex. 675. In Aabc, a = 9, b= 15, and c= 17. Is the triangle obtuse, right, or acute ? Ex. 676. The arms of a right triangle are m2 n2 and 2 mn respectively Find the hypotenuse. Ex. 677. The distance from the center of a chord 14 inches long is 12 inches. Find the radius of the circle. Ex. 678. The distance from the center of a chord 24 inches long is 5 inches. Find the distance from the center of a chord 10 inches long. Ex. 679. In A abc, a = 5, b = 8, and the angle opposite to c equals 60°. Find c. Ex. 680. In A abc, a = 8, b = 15, and the angle opposite to c equals 60°. Find c. Ex. 681. In A abc, a = 3, b=5, and the angle opposite to c equals 120°. Find c. Ex. 682. In A abc, a = 7,5 = 8, and the angle opposite to c equals 120°. Find c. Ex. 683. In A abc, a = 10, b = 17, c = 21. Find the altitude upon 10. Ex. 684. The line of centers of two circles is equal to 30. Find the length of the common chord if the radii are 8 and 26 respectively. Ex. 685. The base of an isosceles triangle is 48 inches. Find the altitude if each arm equals 50 inches. Ex. 686. The diagonal of a square is 20 inches. Find the side. Ex. 687. The sides of a rectangle are 16 and 30 respectively. Find the diagonal. Ex. 688. The diameter AB of a circle is produced to C, and from C a tangent is drawn to the circle. Find the length of the tangent if AB = 30 and BC= 2. Ex. 689. In abc, a = 11, b = 9, and c = 16. Find the median to 16. Ex. 690. In A abc, a = b 18, and c= 22. Find the median to b. Ex. 691. The base of an isosceles triangle is 4, and the arm 7. Find the median to one of the arms. Ex. 692. A ladder 17 ft. long reaches a window 15 ft. high. How far is the lower end of the ladder from the house ? Ex. 693. In A abc, a = 18,5 = 23, and c = 9. Find the bisector of the angle opposite b. Ex. 694. In Aabc, a = 50, b = 32, and c = 78. Find the radius of the circumscribed circle. Ex. 695. The base of an isosceles triangle is b, and each arm a. Find the altitude. Ex. 696. The non-parallel sides AB and CD of a trapezoid are produced till they meet in E. Find AE and BE if AB = 7 and the bases are 5 and 3 respectively. Ex. 697. The altitude of a trapezoid is h, the bases a and b respectively. Find the altitudes of the two triangles formed by producing the non-parallel sides until they meet. Ex. 698. From a point 24 ft. above sea-level the visible horizon has a radius of 6 miles. Find the diameter of the earth. 16, * Ex. 699. In a quadrilateral ABCD AB 15, BC = 20, CD = 16, DA = 18, and AC= 25. Find the diagonal BD. *Ex. 700. Find the greatest segment of a line 10 inches long, when it is divided in extreme and mean ratio. PROBLEMS OF CONSTRUCTION Ex. 701. In a given line AB, to find a point C such that AC:BC =1: Vz. Ex. 702. To divide a given line into segments proportional to any number of given lines by means of Prop. XXIV. Ex. 703. Divide any side of a triangle into two parts proportional to the other two sides. To construct a triangle, having given : (230) Ex. 704. a, b, (b:c). Ex. 705. a, (a:b), and (a:c). Ex. 706. a, b + c, and (b:c). Ex. 707. A, (b:c), tc. Ex. 708. To construct the fourth proportional to three given lines by means of Prop. XXXIV. Ex. 709. From a given rectangle to cut off a similar rectangle by a. line parallel to one of its sides. Ex. 710. In a given circle, to inscribe a triangle similar to a given triangle. Ex. 711. About a given circle, to circumscribe a triangle, similar to a given triangle. Ex. 712. Construct a triangle similar to a given triangle and having a given altitude. Ex. 713. To inscribe a square in a given triangle. (Ex. 610.) Ex. 714. In a given triangle, ABC, to inscribe a parallelogram similar to a given parallelogram, so that one side lies in AB, and the other two vertices lie in BC and AC respectively. (Ex. 610.) Ex. 715. To draw a parallel to one side of a triangle, cutting off another triangle of given perimeter. Ex. 716. From a point without a circle, to draw a secant whose external segment is equal to one-half the secant. Ex. 717. To construct a circle, touching a given circle in a given point, and touching a given line. M Ex. 718. To construct a circle, touching two parallel lines and passing through a given point. Ex. 719. About a given circle, to circumscribe a rhombus, having given an angle. Ex. 720. In a given circle, to inscribe a rectangle, having given the ratio of two sides. *Ex. 721. To divide a trapezoid into two similar trapezoids by a line parallel to the base. Ex. 722. In the prolongation of the side AB of the triangle ABC to find a point X such that AXX BX= (x?. Ex. 723. Through a given point P, to draw a line such that its distances from two other given points, R and S, shall have a given ratio. * Ex. 724. Through a given point within a circle, to draw a chord so that its segments have a given ratio. * Ex. 725. To construct a circle passing through a given point and touching two given lines. *Ex. 726. Through a point of intersection of two circles, to draw a line forming chords which have a given ratio. Ex. 727. To construct two lines, having given their mean proportional and their difference. THEOREMS Ex. 728. If a chord is bisected by another, either segment of the first is a mean proportional between the segments of the other. Ex. 729. If in the triangle ABC the altitudes BD and AE meet in F, and AB = BC, then BC: AF= BD: AD. Ex. 730. Two triangles are similar if an angle of the one is equal to an angle of the other, and the altitudes corresponding with the other angles are proportional. Ex. 731. If, between two parallel tangents, a third tangent is drawn, the radius is the mean proportional between the segments of the third tan nt. Ex. 732. If two circles are tangent externally, and through the point of contact a secant is drawn, the chords formed are proportional to the radii. Ex. 733. If C is the midpoint of the arc AB, and a chord CD meets the chord AB in E, then CE: CA CA: CD. Ex. 734. If two circles intersect, their common chord produced bisects the common tangents. Ex. 735. If an isosceles triangle is inscribed in a circle, the tangents drawn at the vertices form another isosceles triangle. Ex. 736. The tangents drawn at the vertices of an inscribed rectangle enclose a rhombus. Ex. 737. Two parallelograms are similar when they have an angle of the one equal to an angle of the other, and the including sides proportional. Ex. 738. Two rectangles are similar if two adjacent sides are proportional Ex. 739. A circumference described upon the arm of an isosceles triangle as a diameter bisects the base. Ex. 740. Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians. Ex. 741. If in the parallelogram ABCD ZA = 60°, AC2 = AB? + BCP + AB X BC. Ex. 742. If the altitude upon the hypotenuse of a right triangle divides the hypotenuse in extreme and mean ratio, the smaller arm is equal to the non-adjacent segment of the hypotenuse. Ex. 743. If in a triangle the squares of two sides have the same ratio as their projections, upon the third side, the triangle is a right triangle. Ex. 744. If from a point 0, 0A, OB, OC, and OD are drawn so that the angle AOB is equal to the angle BOC, and the angle BOD equal to a right angle, any line intersecting OA, OB, OC, and OD is divided harmonically. (287, 288.) Ex. 745. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the diagonals plus four times the square of the line joining the midpoints of the diagonals. (326.) * Ex. 746. If, from a point within the triangle ABC the perpendiculars OX, OY, and OZ be drawn upon AB, BC, and CA, respectively, AX+ BY? + CZo = BX + YC++ ZA?. *Ex. 747. State and prove the converse of the preceding exercise. *Ex. 748. If two circles are tangent externally, the common tangent is a mean proportional between the diameters. Ex. 749. If through a vertex C of triangle ABC a line CE is drawn parallel to AB, and any point H in CE is joined to F, the midpoint of AB, then FH is divided harmonically by BC and AC produced. (300.) |