NUMERICAL APPLICATIONS OF PLANE

GEOMETRY

METHODS OF NUMERICAL COMPUTATIONS

493. Cancellation. In numerical work in geometry, as elsewhere, the labor of computations may frequently be economized. Those methods of abbreviating work, which are particularly applicable in the ordinary numerical applications of geometry, may be briefly indicated, as follows:

To simplify numerical work by cancellation, group together as a whole all the numerical processes of a given problem, and make all possible cancellations before proceeding to a final numerical reduction.

Ex. Find the ratio of the area of a rectangle, whose base and altitude are 42 and 24 inches, to the area of a trapezoid, whose bases are 21 and 35 and altitude 12.

494. Use of radicals and of л. Where radicals enter in the course of the solution of a numerical problem, it frequently saves labor not to extract the root of the radical till the final answer is to be obtained.

Ex. 1. Find the area of a circle circumscribed about a square whose side is 8.

The diagonal of the square must be 81/2 (Art. 346).

.. the radius of O=41/2.

.. by Art. 449, area of O = T π(4√/2)2=32π=100.6, Area.

Similarly in the use of л, it frequently saves labor not to substitute its numerical value for till late in the process of solution.

Ex. 2. Find the radius of a circle whose area is equal to the sum of the areas of two circles whose radii are 6 and 8 inches, respectively.

Denote the radius of the required circle by x.

Then, by Art. 449, x2 = 36 π +64 π.

π

495. Use of x, y, etc., as symbols for unknown quantities. In some cases a numerical computation is greatly facilitated by the use of a specific symbol for an unknown quantity.

Ex. In a triangle whose sides are 12, 18, and 25, find the segments of the side 25 made by the bisector of the angle opposite.

Denote the required segments of side 25 by

496. Limitations of numerical computations. Owing to the limitations of human eyesight and of the instruments used in making measurements, no measurement can be accurate beyond the fifth or sixth figure; and in ordinary work, such as is done by a carpenter, measurements are not accurate beyond the third figure. As all numerical applications of geometry are based on practical measurements, it is not necessary to carry arithmetical work beyond the fifth or sixth significant digit.

Other methods of facilitating numerical computations, as by the use of logarithms, are beyond the scope of this book.

T

NUMERICAL PROPERTIES OF LINES

EXERCISES. GROUP 53

THE RIGHT TRIANGLE

Ex. 1. Find the hypotenuse of a right triangle whose legs are 12 and 35.

Ex. 2. The hypotenuse of a right triangle is 29, and one leg is 20. Find the other leg.

Ex. 3. If a window is 15 ft. from the ground and the foot of a ladder is to be 8 feet from the house, how long a ladder is necessary to reach the window?

Ex. 4. Find the diagonals of a rectangle whose sides are 5 and 12.

Ex. 5. If the base of an isosceles triangle is 8 and a leg is 5, find the altitude.

Ex. 6. Find the diagonal of a square whose side is 1 ft. 6 in.

Ex. 7. The diagonals of a rhombus are 24 and 10. Find a side.

Ex. 8. One side of a rhombus is 17 and one diagonal is 30. Find the other diagonal.

Ex. 9. In a circle whose radius is 5, find the length of the longest and shortest chords through a point at a distance 3 from the center.

Ex. 10. In a circle whose radius is 25 in., find the distance from the center to a chord 48 in. long.

Ex. 11. If a chord 12 in. long is 5 in. from the center of a circle, how far from the center of the same circle is a chord 10 in. long?

Ex. 12. A ladder 40 ft. long reaches a window 20 ft. high on one side of a street and, if turned on its foot, reaches a window 30 ft. high on the other side. How wide is the street?

Ex. 13. If one leg of a right triangle is 10 and the hypotenuse is twice the other leg, find the hypotenuse.

Ex. 14. Find the altitude of an equilateral triangle whose side is 6.

Ex. 15. Find the side of an equilateral triangle whose altitude is 8.

Ex. 16. Find the side of a square whose diagonal is 15.

Ex. 17. One leg of a right triangle is 3, and the sum of the hypotenuse and the other leg is 9. Find the sides.

Ex. 18. A tree 90 ft. high is broken off 40 ft. from the ground. How far from the foot of the tree will the top strike?

Ex. 19. The radii of two circles are 1 and 6 in., and their centers are 13 in. apart. Find the length of the common external tangent.

Ex. 20. The sides of a triangle are 10, 11, 12. Find the length of the projection of the side whose length is 10, on the side 12.

EXERCISES. GROUP 54

TRIANGLES IN GENERAL

Ex. 1. The sides of a triangle are 12, 18 and 20. Find the segments of the side 20, made by the bisector of the angle opposite.

Ex. 2. In the same triangle, find the segments of the side 20, made by the bisector of the exterior angle opposite.

Ex. 3. If the legs of a right triangle are 6 and 8, find the hypotenuse, the altitude on the hypotenuse, and the projections of the legs on the hypotenuse.

Ex. 4. Is a triangle acute, obtuse, or right, if the three sides are 5, 12, 14; 5, 11, 12; 5, 12, 13; 4, 5, 6?

Ex. 5. If the sides of a triangle are 6, 7 and 8, compute the length of the altitude on 8.

Ex. 6. Also the length of the median on the same side.

Ex. 7. Also the length of the bisector of the angle opposite the side 8.

Ex 8. If two sides and a diagonal of a parallelogram are 8, 12 and 10, find the other diagonal.

[SUG. Use Art. 352.]

Ex. 9. Two sides of a triangle are 10 and 18, and the median to the third side is 12. Find the third side.

Ex. 10. Two sides of a triangle are 17 and 16, and the altitude on the third side is 15. Find the third side.

Ex. 11. The hypotenuse of a right triangle is 10, and the altitude on the hypotenuse is 4. Find the segments of the hypotenuse and the legs.

Ex. 12. Find the three medians, the three bisectors, and the three altitudes of a triangle whose sides are 13, 14, 15.

Using T

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EXERCISES. GROUP 55

CIRCUMFERENCES AND ARCS

Ex. 1. Find the circumference of a circle whose radius is 1 ft. 9 in. Ex. 2. Find the radius of a circle whose circumference is 121 ft.

Ex. 3. A bicycle wheel 28 in. in diameter makes, in an afternoon, 3,000 revolutions. How far does the bicycle travel?

Ex. 4. What is the diameter of a wheel which makes 1,400 revolutions in going 8,800 yds. ?

Ex. 5. If the diameter of a circle is 20, find the length of an arc of 60°; also of 83°.

Ex. 6. If the length of an arc is 14 and the radius is 6, find the number of degrees in the arc.

Ex. 7. If the arc of a quadrant is 1 ft. in length, find the diameter.

Ex. 8. Two concentric circumferences are 88 and 132 in. in length, respectively. Find the width of the circular ring between them.

Ex. 9. If the year be taken as 365 da., and the earth's orbit a circle whose radius is 93,250,000 miles, find the velocity of the earth in its orbit per second.