366. In any quadrilateral the sum of the squares of the four sides is equal to the sum of the squares of the diagonals, plus four times the square of the line joining the middle points of the diagonals. Sug. Use 365. What modification of the above theorem would result if the quadrilateral were a parallelogram? 367. If from any point, selected at random, in the circumference of a circle, a perpendicular be drawn to a diameter, this perpendicular will be a mean proportional between the two segments of the diameter. 368. If two chords of a circle intersect each other, the four parts form a proportion. 369. If two secants be drawn from the same point without the circle, the entire secants and the parts that are without the circle form a proportion. 370. If, from the same point, a tangent and secant to a circle be drawn, the tangent, the secant, and that part of the latter that is outside the circle, will form a proportion. 371. The two tangents to two intersecting circles from any point in their common secant are equal. Sug. Consult 370. 372. If two circles intersect each other, their common chord extended bisects their common tangents. ADVANCE THEOREMS. 373. In any triangle the product of any two sides is equal to the diameter of the circumscribed circle multiplied by the perpendicular drawn to the third side from the vertex of the angle opposite. Sug. Construct the diameter from the same vertex as the perpendicular, and join its extremity with one of the other vertices, making a right triangle similar to the one formed by the perpendicular, whence the necessary proportion. 374. In any triangle the product of any two sides is equal to the product of the segments of the third side, formed by the bisector of the opposite angle, plus the square of the bisector. Sug. Circumscribe a circle, and extend the bisector to the circumference, and connect its extremity with one of the other vertices of the triangle. Then if the vertices of the triangle be lettered A, B, and C, and the bisector be drawn from A, and D the point where the bisector crosses the side BC, and E the extremity of the bisector extended, then the triangles BAD and ACE can be proved similar. 375. If one side of a right triangle is double the other, the perpendicular, from the vertex of the right angle to the hypothenuse, divides it into segments which are to each other as 1 to 4. 376. A line parallel to the bases of a trapezoid, passing through the intersection of the diagonals, and terminating in the non-parallel sides, is bisected by the diagonals. 377. In any triangle the product of any two sides is equal to the product of the segments of the third side, formed by the bisector of the exterior angle at the opposite vertex, minus the square of the bisector. Sug. Consult Theorem 374. 378. The perpendicular, from the intersection of the medians of a triangle, upon any straight line in the plane of the triangle, is one-third the sum of the perpendiculars from the vertices of the triangle upon the same line. 379. If two circles are tangent to each other, their common tangent and their diameters form a proportion. 380. 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. 381. In any quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of the opposite sides. Sug. From one vertex draw a line to the opposite diagonal, making the angle formed by it and one side equal to the angle formed by the other diagonal and side which meets the former. 382. If three circles whose centres are not in the same straight line intersect one another, the common chords will intersect each other at one point. 383. If two chords be perpendicular to each other, the sum of the squares of the four segments is equal to the square of the diameter. 384. The sum of the squares of the diagonals of a quadrilateral is equal to twice the sum of the squares of the lines joining the middle points of the opposite sides. PROBLEMS OF COMPUTATION. 385. The chord of one-half a certain arc is 9 inches, and the distance from the middle point of this arc to the middle of its subtending chord is 3 inches. Find the diameter of the circle. 386. The external segments of two secants to a circle from the same point are 10 inches and 6 inches, while the internal segment of the former is 5 inches. What is the internal segment of the latter? 387. The hypothenuse of a right triangle is 16 feet, and the perpendicular to it from the vertex of the opposite angle is 5 feet. Find the values of the legs and the segments of the hypothenuse. 388. The sides of a certain triangle are 6, 7, and 8 feet respectively. In a similar triangle the side corresponding to 8 is 40. Find the other two sides. 389. The sides of a certain triangle are 9, 12, and 15 feet respectively. Find the segments of the sides made by the bisectors of the several angles. 390. If a vertical rod 6 feet high cast a shadow 4 feet long, how high is the tree which, at the same time and place, casts a shadow 90 feet long? 391. The perimeters of two similar polygons are 200 and 300 feet respectively, and one side of the former is 24 feet. What is the corresponding side of the latter? 392. How long must a ladder be to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house? 393. Find the lengths of the longest and the shortest chord that can be drawn through a point 6 inches from the centre of a circle whose radius is 10 inches. 394. The distance from the centre of a circle to a chord 10 inches long is 12 inches. Find the distance from the centre to a chord 24 inches long. 395. The radius of a circle is 5 inches. Through a point 3 inches from the centre a diameter is drawn, and also a chord perpendicular to the diameter. Find the length of this chord, and the distance from one end of the chord to the ends of the diameter. 396. Through a point 10 feet from the whose radius is 6 feet tangents are drawn. centre of a circle Find the lengths of the tangents and of the chord joining the points of contact. 397. If a chord 8 feet long be 3 feet from the centre of the circle, find the radius and the distances from the end of the chord to the ends of the diameter which bisects the chord. 398. Through a point 5 inches from the centre of a circle whose radius is 13 inches any chord is drawn. What is the product of the two segments of the chord? What is the length of the shortest chord that can be drawn through that point? 399. From the end of a tangent 20 inches long a secant is drawn through the centre of a circle. If the exterior segment of this secant be 8 inches, what is the radius of the circle? 400. A tangent 12 feet long is drawn to a circle whose radius is 9 feet. Find the external segment of a secant through the centre from the extremity of the tangent. 401. The span of a roof is 28 feet, and each of its slopes. measures 17 feet from the ridge to the eaves. Find the height of the ridge above the eaves. 402. A ladder 40 feet long is placed so as to reach a window 24 feet high on one side of the street, and on turning the ladder over to the other side of the street, it just reaches a window 32 feet high. What is the width of the street? 403. The bottom of a ladder is placed at a point 14 feet from a house, while its top rests against the house 48 feet from the ground. On turning the ladder over to the other side of the street its top rests 40 feet from the ground. the width of the street. Find 404. One leg of a right triangle is 3925 feet, and the difference between the hypothenuse and other leg is 625 feet. Find the hypothenuse and the other leg. AREAS. 405. The area of a surface is its numerical measure; i.e. the numerical expression for the number of times it contains another surface arbitrarily assumed as a unit of measure. For example, the area of the floor of a room is the number of times it contains some one of the common units of surface, square foot, square yard, etc. This unit of surface is called the superficial unit. The most convenient superficial unit is |