340. 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. Ex. 991. In ▲ abc, a = 6, b = 10, and he = 4; find the diameter of the circumscribed circle. Ex. 992. In ▲ abc, a = 10, b = 15, h2 = 6. Find the radius of the circumscribed circle. Ex. 993. ▲ ABC is inscribed in a circle of radius = 5 inches. Find the altitude to BC if AB = 4, and AC = 5. Ex. 994. Find the diameter of the circle circumscribed about ▲ abc, if (a) a = 17, b = 8, c = 15. = 21. (b) a = 10, b = 17, c = Ex. 995. In ▲ abc, a = 20, b = 15, and the projection of b upon c equals 9. Find the radius of the circumscribed circle. Ex. 996. In A abc, a = 9 and b = 12. Find c if the diameter of the circumscribed circle equals 15. [See practical applications, p. 300.] PROBLEMS OF COMPUTATION Ex. 997. The arms of a right triangle are 8 and 15 respectively. Compute the hypotenuse and the altitude upon the hypotenuse. Ex. 998. In A abc, a = 9, b = 15, and c = 17. right, or acute? Ex. 999. The arms of right triangle are m2 tively. Find the hypotenuse. Is the triangle obtuse, n2 and 2 mn respec Ex. 1000. A chord 14 in. long is 12 in. distant from the center. Find the radius of the circle. Ex. 1001. A chord 24 in. long is 5 in. distant from the center. Find the distance of the center from a chord 10 in. long. Ex. 1002. In Δ abc, α 60°. Find c. Ex. 1003. 5:4. A straight line AB = 4, is divided externally in the ratio Find the segments. Ex. 1004. The shadow of a church steeple upon level ground is 60 ft., while a pole 10 ft. high casts a shadow 2 ft. long. How high is the steeple ? Ex. 1005. Find the product of the segments of a chord drawn through a point 8 in. from the center of a circle whose radius is 10 in. What is the length of the shortest chord that can be drawn through that point? Ex. 1006. In ▲ abc, b = 13, c = 35, and the angle opposite to a equals 120°. Find a. Ex. 1007. In ▲ abc, a = 10, b=17, c=21. Find the altitude upon 10. Ex. 1008. The line of centers of two circles is equal to 10. Find the length of the common chord if the radii are 8 and 6 respectively. Ex. 1009. The base of an isosceles triangle is 48 in. Find the altitude if each arm equals 50 in. Ex. 1010. Two sides and a diagonal of a parallelogram are 7, 9, and 8 respectively; find the length of the other diagonal. Ex. 1011. The diagonal of a square is 20 in. Find the side. Ex. 1012. The sides of a rectangle are 16 and 30 respectively. Find the diagonal. Ex. 1013. a tangent is The diameter AB of a circle is produced to C, and from C drawn to the circle. Find the length of the tangent if AB = 30 and BC= 2. Ex. 1014. In ▲ abc, a = 16, b=18, and c = 22. Find the median to b. The base of an isosceles triangle is 4, and the arm 7. Find the median to one of the arms. Ex. 1015. Ex. 1016. A ladder 17 ft. long reaches a window 15 ft. high. How far is the lower end of the ladder from the house? Ex. 1017. In ▲ abc, a = 18, b = 23, and c = 9. Find the bisector of the angle opposite b. Ex. 1018. In ▲ abc, c = 10, ZA = 30°, and ≤ C = 90°. Find a and b. Ex. 1019. The diagonals of a parallelogram are 30 and 26 in. Find the altitude if the base equals 14 in. Ex. 1020. In ▲ abc, a = 17, b circumscribed circle. = 10, c9. Find the radius of the Ex. 1021. The base of an isosceles triangle is b, and each arm a. Find the altitude. Ex. 1022. 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. 1023. 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. 1024. From a point 24 ft. above sea level the visible horizon has a radius of 6 miles. Find the diameter of the earth. Ex. 1025. Find the length of the common internal tangent of two circles if the line of centers is 17, and the radii are 5 and 3 respectively. Ex. 1026. A chord 30 in. long subtends an angle of 120°. Find the distance of the chord from the center of the circle. * Ex. 1027. In triangle ABC, AB = BC = 25, AC = 30, and on AB is laid off AD = 8. Find the length of CD. * Ex. 1028. The three sides of a triangle are 14, 16, and 6. Find the angle opposite 14. * Ex. 1029. In a quadrilateral ABCD, AB = 10, BC = 17, CD = 13, DA 20, and AC 21. Find the diagonal BD. = PROBLEMS OF CONSTRUCTION Ex. 1030. Divide any side of a triangle externally into two parts proportional to the other two sides. To construct a triangle, having given : Ex. 1031. a, b, and b c = 4:5. Ex. 1032. Ex. 1033. (230) From a given rectangle to cut off a similar rectangle by a line parallel to one of its sides. Ex. 1034. In a given circle, to inscribe a triangle similar to a given triangle. Ex. 1035. About a given circle, to circumscribe a triangle, similar to a given triangle. Ex. 1036. Construct a triangle similar to a given triangle and having a given altitude. Ex. 1037. To inscribe a square in a given triangle. Ex. 1038. Assuming an arbitrary unit construct a line equal to (a) √2, (b) √3, (c) 1+ v5. Ex. 1039. Construct a line that shall be to a given line as 1: √2. Ex. 1040. Construct a line that shall be to a given line as √3: 1. * Ex. 1041. Construct a triangle similar to a given triangle and having a given median. Ex. 1042. In a given line AB, to find a point C such that AC : BC = 1: √2. Ex. 1043. To construct a parallelogram similar to a given parallelogram and having a given diagonal. Ex. 1044. To construct a triangle similar to a given triangle and having a given perimeter. Ex. 1045. If a and b are two given lines, construct a line equal to b Ex. 1046. From a point without a circle, to draw a secant whose external segment is equal to one half the secant. Ex. 1047. To construct a circle, touching a given circle in a given point, and touching a given line. HINT. Draw a tangent to the circle. Ex. 1048. To construct a circle, touching two parallel lines and passing through a given point. Ex. 1049. In a given circle, to inscribe a rectangle, having given the ratio of two sides. *Ex. 1050. To divide a trapezoid into two similar trapezoids by a line parallel to the base. Ex. 1051. In the prolongation of the side AB of the triangle ABC to find a point X such that AX × BX = CX2. (322.) Ex. 1052. 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. 1053. Through a given point within a circle, to draw a chord so that its segments have a given ratio. 1 * Ex. 1054. Through a point of intersection of two circles, to draw a line forming equal chords. * Ex. 1055. To construct two lines, having given their mean proportional and their difference. THEOREMS Ex. 1056. If a chord is bisected by another, either segment of the first is a mean proportional between the segments of the other. Ex. 1057. If in the triangle ABC the altitudes BD and AE meet in F, and AB = BC, then Ex. 1058. 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. 1059. If between two parallel tangents, a third tangent is drawn, the radius is the mean proportional between the segments of the third tangent. Ex. 1060. 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. 1061. If C is the mid-point of the arc AB, and a chord CD meets the chord AB in E, then of the inscribed circle is equal to the sum of the arms of the triangle. Ex. 1064. If two circles intersect, their common chord produced bisects the common tangents. Ex. 1065. If an isosceles triangle is inscribed in a circle, the tangents drawn at the vertices form another isosceles triangle. Ex. 1066. The tangents drawn at the vertices of an inscribed rectangle inclose a rhombus. Ex. 1067. 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. 1068. Two rectangles are similar if two adjacent sides are proportional. Ex. 1069. 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. 1070. If in rectangle ABCD a perpendicular is drawn from D upon AC, the prolongation of which intersects AB in E, then AE: AD AD: CD. Ex. 1071. Two circles are tangent externally, and through the point of contact two straight lines are drawn terminating in the circumferences; prove that the corresponding segments of the lines are proportional. Ex. 1072. If two circles intersect in A and B and the two chords BD and BC are respectively tangents to the two circles, AB is the mean proportional between AD and AC. B |