7. The sum of the three perpendiculars from a point within an equilateral triangle to the three sides is equal to the altitude of the triangle. 8. The bases of two equivalent triangles are 10 ft. and 15 ft. respecFind the ratio of their altitudes.

tively.

12. A square is greater than any other rectangle inscribed in the same

circle.

[Show that both square and rectangle have diameters for diagonals.]

16. Draw a line through the point of intersection of the diagonals of a trapezoid dividing it into two equivalent trapezoids.

17. The square described on the sum of two lines is equivalent to the sum of the squares of the lines increased by twice their rectangle.

18. The square described on the difference of two lines is equivalent to the sum of the squares of the lines diminished by twice their rectangle.

19. The rectangle having for its sides the sum and the difference of two lines is equivalent to the difference of their squares.

20. A triangle and a rectangle having equal bases are equivalent. How do their altitudes compare?

21. Draw a straight line through a vertex of a triangle dividing it into two parts having the ratio of m to n.

22. Through a given point within or without a parallelogram draw a line dividing the parallelogram into two equivalent parts.

=

ab

23. If a and b are the sides of a triangle, show that its area when the included angle is 30° or 150°; ab √2 when the included angle is 45° or 135°; ab √3 when the included angle is 60° or 120°. [Using either a or b for base, find the altitude of the A.]

24. If equilateral triangles are described on the three sides of a rightangled triangle, prove that the triangle on the hypotenuse is equivalent to the sum of the triangles on the other sides.

25. On a given line as a base construct a rectangle equivalent to a given rhombus.

26. Bisect a triangle by a line drawn parallel to one of its sides. [$ 616.]

27. The square of a line from the vertex of an isosceles triangle to the base is equivalent to the square of one of the equal sides diminished by the rectangle of the segments of the base [i.e. BD2 = AB2 · AD × DC].

the altitude to AC. Use § 643.]

[Draw

28. If, in Exercise 27, BD is drawn to a

A

B

point D on the prolonged base, then_BD2 = AB2 + AD × DC.

29. Three times the sum of the squares on the sides of a triangle is equivalent to four times the sum of the squares on its medians. [$ 665.]

30. If the base a of a triangle is increased d inches, how much must the altitude b be diminished in order that the area of the triangle shall be unaltered.

31. OC is a line drawn from the center of the circle to any point of the chord AB.

Prove that OC2 = ŌA2 – AC × CB.

32. The lengths of the parallel sides of a trapezoid are a ft. and b ft. respectively. The two inclined sides are each c ft. Find the area of the trapezoid.

33. From the middle point D of the base of the right-angled triangle ABC, DE is drawn perpendicular to the hypotenuse BC.

Prove that BE2 – EC2 = AB2.

B

B

E

34. In any circle the sun of the squares on the segments of two chords that are perpendicular to each other is equivalent to the square on the diameter. [§ 643.]

35. Construct a triangle having given its angles and its area.

B

36. In the triangle ABC, AD, BE, and CF are lines drawn from the vertices and passing through a common point 0.

F

38. Given the altitude, one of the angles, and the area, construct

a parallelogram.

39. The two medial lines AE and CD of the triangle ABC intersect at F. Prove the triangle AFC equivalent to the quadrilateral BDFE.

40. The diagonals of a trapezoid divide it into four triangles, two of which are similar, A and the other two equivalent.

41. Any two points, C and D, in the semicircumference ACB are joined with the extremities of the diameter AB. AE and BF are drawn perpendicular to the chord DC prolonged.

Prove that CE2 + CF2 = DE2 + DF2. [Use § 643.]

A

D

E

B

42. Describe four circles each of which is tangent to three lines that form a triangle.

[One of the four is the inscribed circle of the A, and its radius is denoted by r. The other three are called escribed circles of the triangle, and their radii are denoted (ra is the radius

by ra, rb, and re

are also expressions for the area of triangle ABC.

44. The area of triangle ABC = √r × ra × rb × rc. [Ex. 43.]

45. Prove that ra + rb + re — r = 4 R [R radius of the circle circumscribed about ▲ ABC]. [Ex. 43 and § 689.]