15. Show how to find, by means of similar triangles, the distance AX (Fig. 63). If AP=200 feet, OP 20 feet, OQ=32 feet, find AX.

16. Show how to find the distance between two inaccessible objects X, Y. (Fig. 64.) If AX=4 miles, AY 5 miles, AB = 200 feet, AC=250 feet, BC=225 feet, find XY.

17. The perimeters of two similar polygons are 280 feet and 160 feet. If a side of the first polygon is 15 feet, find the homologous side of the second.

18. Required the length of a ladder which will reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the building.

19. How far apart are the opposite corners of a floor 12 feet by 16 feet?

20. If the side of an equilateral triangle is a, find the altitude.

21. Find the legs of a right triangle if their projections upon the hypotenuse are 2.88 feet and 5.12 feet.

x2: y2 = 2.88: 5.12, and x2 + y2 = 64.

22. The legs of a right triangle are 10 feet and 24 feet. Find their projections upon the hypotenuse, and the altitude upon the hypotenuse.

23. Given the legs a, b of a right triangle, find their projections x, y on the hypotenuse, and the altitude h upon the hypotenuse.

24. The hypotenuse of a right triangle is 1, and the sum of the legs is 1.4; find the legs.

25. Find the three sides of a right triangle if these sides are three consecutive integral numbers.

26. Compute the legs of a right triangle if their ratio is 34, and the hypotenuse is 40.

27. Find the sides of a right triangle if their sum is 132 and the sum of their squares is 6050.

28. If in a right triangle the hypotenuse is 25 feet, one leg is 15 feet; find the altitude upon the hypotenuse.

29. The legs of a right triangle are 3.128 and 4.275; compute to 0.001 of a unit the segments of the hypotenuse made by the bisector of the right angle.

30. The radius of a circle is 5 inches. Find the distance from the centre to a chord 8 inches long.

31. The radius of a circle is r. What is the length of a chord the distance of which from the centre is r? What angle does it subtend at the centre?

32. If the radii of two concentric circles are 10 inches and 8 inches, find the length of a chord in the larger circle which touches the smaller circle.

33. Two circles, the radii of which are r, r', intersect at right angles; find the distance between their centres.

The tangents at the point of intersection are perpendicular to each other, and form with the line of centres a right triangle.

34. The radii of two circles are 8 inches and 3 inches, and the distance between their centres is 15 inches. Find the length of their common exterior tangent.

35. The radius of a circle is 6 inches. Through a point 10 inches from the centre, tangents are drawn. Find the lengths of these tangents and of the chord of contact.

36. The distance between two parallel lines is a, and the distance between two points A, B in one of the lines is 26. Find the radius of a circle which passes through A and B, and touches the other line. What is its value if a b?

=

37. The sides of a triangle are 8, 9, 13; is the greatest angle acute, right, or obtuse?

38. If in an isosceles triangle a denote one of the equal sides, and b half the base, find the radius of the circumscribed circle.

Draw the altitude, and drop a perpendicular from the centre of the circle to one of the equal sides; two similar triangles are formed.

39. In an isosceles trapezoid let a = the greater base, b the other base, c one of the legs; find the lengths of b=

the diagonals.

The two diagonals are equal. (Ex. 51, § 2.) Draw (Fig. 65) CE I to DA, CFL to AB; in the isos. A CBE, FB = ¦ EB = (a − b) Then apply No. 162 to the ▲ ACB.

40. Compute the sides of a rectangle, given a diagonal d and the perimeter 2p. When is the problem possible? when impossible?

Let x and y denote the two sides; then x + y = p, x2 + y2 = d2 ; whence,

x = P + √ 2 ď2 — p2

2

If p2<2d2, or p<d√2, the problem is possible; if p2>2d2, or p>d√2, the roots are imaginary, and the problem is impossible. If p = d√2, then x = y = p, and the rectangle is a square.

41. Two chords AB, CD intersect in M; if AM=5 inches, BM= 6 inches, CD=11.5 inches, find CM and MD.

x+y=11.5, xy = 30.

=

4

42. Two chords AB, CD intersect in M; if AM: inches, BM= 5 inches, and the difference between CM and MD-8 inches, find CD.

43. The diameter of a circle is equal to 30 feet, and is divided into three equal parts; find the lengths of the perpendiculars drawn from the points of division and terminated by the circumference.

44. What must be the distance of a point from the centre of a circle (radius=r) in order that a tangent drawn from a point to the circle may be equal to three times the radius?

45. Through a point P, exterior to a circle, a tangent PA and a secant PBC are drawn; if PB = 5 inches, BC 4 inches, find PA.

46. Find in a line AB touching a circle (radius = r) in A, a point C, such that the exterior part CD of the line which joins Cto the centre shall be equal to AC.

47. The radius of a circle equals 2 inches. Through a point A, 4 inches from the centre, a secant ABC is drawn. If BC=1 inch, find AB.

3 or

Let x = AB (Fig. 67); then x(x + 1) = 12, whence x 4. The negative root must be rejected, or else considered as the solution of the question obtained by changing x to - a in the equation x(x+1) 12. The equation then becomes x2 — x = 12, and belongs to the question, find the length of the secant AC if BC= 1 in.

=

48. In a triangle given the sides, a = 1551 feet, b = 2068 feet, c=2585 feet; find the median to the side a.

+b2 = 2x2 + 1⁄2 c2.

If x equal the required line, then (Ex. 19, ? 13) x2 In this particular case the labor of computing a may be avoided by observing that the given numbers are equimultiples of 3, 4, 5, respectively; therefore the triangle is a right triangle (No. 161), and xc. (Ex. 45, 2.)

49. In a triangle given the sides a, b, c; find the lengths of the three medians. (Ex. 19, § 13.)

By adding the squares of these values, we obtain

x2 + y2 + z2 = 3 (a2 + b2 + c2) ;

or, the sum of the squares of the medians is equal to three-fourths of the sum of the squares of the sides.