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Remark.-It may here be observed that the above expression for the diameter of the circumscribing circle is symmetrical with respect to the sides, as might have been expected.

PROP. VI.-To divide a line into extreme and mean ratio.

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1. Find the hypothenuse of a right-angled triangle of which the base and perpendicular are :

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2. Find the base of a right-angled triangle of which

the hypothenuse and perpendicular are:

(1.) 200 and 112.

(2.) 12.5 and 6.25.

(3.) 17.2 and 16.1.

3. Find the perpendicular of a right-angled triangle

of which the hypothenuse and base are :

(1.) 100 and 50.

1.25 and .05.

(3.) 500 and 300.

4. Find the diagonal of a rectangle whose adjacent sides are 15 and 12.

5. Find the perimeter of a right-angled triangle of which the base is 36 and the hypothenuse is 60.

6. The diagonal of a rectangle is 100, and one of the sides 50; find the other side.

7. The side of a square is 5 feet; find its diagonal. 8. The diagonal of a square is 100; find its side. 9. Find the perimeter of a square whose diagonal is 12.5 feet.

10. Find the perimeter of a square inscribed in a circle whose diameter is 1.

11. Find the radius of the circle circumscribing a square whose side is 6 feet.

12. Find the diagonal of a cube, the edge of which is 12 feet.

13. The diagonal of a cube is 15 feet; find its edge.

14. Find the distance from the centre of a circle of a chord whose length is 30 feet in a circle whose radius is 20.

15. The radius of a circle is 15 inches, the perpendicular from the centre on a chord is 12 inches; find the length of the chord.

16. Find the length of the least chord, drawn through a point 3 inches from the centre of a circle whose diameter is 10 inches.

17. A ladder 50 feet long reaches a point in a wall 30 feet high; how far is the foot of the ladder from the bottom of the wall?

18. A ladder 50 feet long reaches a point in a wall 30 feet high; the foot of the ladder is drawn out along the ground 3 feet; how far will the top of the ladder slip down the wall?

19. A pole 60 feet high is broken by the wind, and the top strikes the ground 10 feet from the foot of the pole; how far from the ground is the pole broken?

20. In an equilateral triangle, whose side is 10 feet, find the length of the perpendicular from any of the angles on the opposite side.

21. The perpendicular height of an equilateral triangle is p; find the length of its side.

22. The difference between the diagonal and side of a square is 5 feet; find the length of the side.

23. The sum of the diagonal and side of a square is 5 feet; find the side.

24. Find the perimeter of an isosceles triangle, each of whose equal sides is s, and perpendicular on base p. 25. Find the lengths of the perpendiculars on the longest and shortest sides in the triangles whose sides

are:

(1.) 200, 250, 300. (2.) 10, 12, 20. (3.) 17, 18, 19.

(4.) 3, 4, 5.

(5.) 6, 8, 10.

(6.) 5, 10, 10.

26. Shew how to find the segments of the sides in each of the above cases.

27. Find also the diameters of the circumscribing circles.

28. Find the lengths of the lines drawn to the middle of the greatest and least sides from the opposite angles in the triangles whose sides are:

(1.) 200, 300, and 350.

(2.) 500, 600, and 750.

29. Find also the lengths of the lines bisecting the angle between the greatest and least side, in each of the above triangles.

30. The base of a triangle is 48 feet, the height 20 feet, and one of the sides 24 feet; find the other side. 31. In a right-angled triangle, having given the sides containing the right angle equal to a and b, find

(1.) The perpendicular from the right angle on the hypothenuse.

(2.) The segments into which the perpendicular divides the hypothenuse.

32. In a triangle ABC, if Aa be drawn from angle A to the middle of the opposite side, and Bb from B to the middle of the opposite side, intersecting each other in O; prove

aO

=

aA.

B

or

THE CIRCLE.

PROP. VII.-Having given the height of an arc, and the radius of the circle, to find the length of the chord.

H

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Interpreting this formula, we have the following rule for determining the chord: Multiply twice the radius by the height; from the product subtract the square of the height, and take twice the square root of the remainder.

Remark.-It may here be noticed that, in the above formula,

when h=0, or 2r, the chord is zero;

and when h=r, the chord becomes the diameter.

Cor. The above formula involves the three quantities, l, r, and h, and expresses the value of in terms of r and h; but it also enables us to express any one of the three quantities in terms of the other two.

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Interpreting this, we have the following rule for finding the radius: Add the square of the chord to four times the square of the height, and divide the sum by eight times. the height.

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This will be readily

H

B

The two roots of this equation shew that we have two values of h for each value of 7. understood by referring to the geometrical figure; for if we inscribe a chord AB equal to 7, it is evident that we have another chord A'B' also equal to 7, parallel to AB, and equally distant from the centre; the two heights which correspond to a chord equal to are CD and CD'. We see also that CD + CD' = 2r,

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A

B

PROP. VIII.—Having given the side of a regular polygon inscribed in a given circle, to find the side of the regular polygon inscribed with double the

number of sides.

Let AB=7, the side of the given inscribed polygon.

AO=r. CH being the diameter perpendicular to AB through its middle point D, AC will be the side required, and which we will denote by k. We evidently have 2r: k::k: CD;

A

B

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