EXERCISE 55

PROBLEMS OF COMPUTATION

1. The sides of a triangle are 0.7 in., 0.6 in., and 0.7 in. respectively. Is the largest angle acute, right, or obtuse?

2. The sides of a triangle are 5.1 in., 6.8 in., and 8.5 in. respectively. Is the largest angle acute, right, or obtuse?

3. Find the area of an isosceles triangle whose perimeter is 14 in. and base 4 in. (One decimal place.)

4. Find the area of an equilateral triangle whose perimeter is 18 in. (One decimal place.)

5. Find the area of a right triangle, the hypotenuse being 1.7 in. and one of the other sides being 0.8 in.

6. Find the ratio of the altitudes of two triangles of equal area, the base of one being 1.5 in. and that of the other 4.5 in.

7. The bases of a trapezoid are 34 in. and 30 in., and the altitude is 2 in. Find the side of a square having the same area.

8. What is the area of the isosceles right triangle in which the hypotenuse is √2?

9. What is the area of the isosceles right triangle in which the hypotenuse is 7√2?

10. If the side of an equilateral triangle is 2√3, what is the altitude of the triangle? the area of the triangle?

11. If the side of an equilateral triangle is 1 ft., what is the area of the triangle?

12. If the area of an equilateral triangle is 43.3 sq. in., what is the base of the triangle? (Take √3 = 1.732.)

13. The sides of a triangle are 2.8 in., 3.5 in., and 2.1 in. respectively. Draw the figure carefully and see what kind of a triangle it is. Verify this conclusion by applying a geometric test, and find the area of the triangle.

EXERCISE 56

THEOREMS

1. The area of a rhombus is equal to half the product of

its diagonals.

2. Two triangles are equivalent if the base of the first is equal to half the altitude of the second, and the altitude of the first is equal to twice the base of the second.

3. The area of a circumscribed polygon is equal to half the product of its perimeter by the radius of the inscribed circle.

4. Two parallelograms are equivalent if their altitudes are reciprocally proportional to their bases.

5. If equilateral triangles are constructed on the sides of a right triangle, the triangle on the hypotenuse is equivalent to the sum of the triangles on the other two sides.

6. If similar polygons are constructed on the sides of a right triangle, as corresponding sides, the polygon on the hypotenuse is equivalent to the sum of the polygons on the other two sides.

Ex. 6 is one of the general forms of the Pythagorean Theorem.

7. If lines are drawn from any point within a parallelogram to the four vertices, the sum of either pair of triangles with parallel bases is equivalent to the sum of the other pair.

8. Every line drawn through the intersection of the diagonals of a parallelogram bisects the parallelogram.

9. The line that bisects the bases of a trapezoid divides the trapezoid into two equivalent parts.

10. If a quadrilateral with two sides parallel is bisected by either diagonal, the quadrilateral is a parallelogram.

11. The triangle formed by two lines drawn from the midpoint of either of the nonparallel sides of a trapezoid to the opposite vertices is equivalent to half the trapezoid.

EXERCISE 57

PROBLEMS OF CONSTRUCTION

1. Given a square, to construct a square of half its area. 2. To construct a right triangle equivalent to a given oblique triangle.

3. To construct a triangle equivalent to the sum of two given triangles.

4. To construct a triangle equivalent to a given triangle, and having one side equal to a given line.

5. To construct a rectangle equivalent to a given parallelogram, and having its altitude equal to a given line.

6. To construct a right triangle equivalent to a given triangle, and having one of the sides of the right angle equal to a given line.

7. To construct a right triangle equivalent to a given triangle, and having its hypotenuse equal to a given line.

8. To divide a given triangle into two equivalent parts by a line through a given point P in the base.

9. To draw from a given point P in the base AB of a triangle ABC a line to AC produced, so that it may be bisected by BC.

10. To find a point within a given triangle such that the lines from this point to the vertices shall divide the triangle into three equivalent triangles.

11. To divide a given triangle into two equivalent parts by a line parallel to one of the sides.

12. Through a given point to draw a line so that the segments intercepted between the point and perpendiculars drawn to the line from two other given points may have a given ratio.

13. To find a point such that the perpendiculars from it to the sides of a given triangle shall be in the ratio p, q, r.

EXERCISE 58

REVIEW QUESTIONS

1. What is meant by the area of a surface? Illustrate.

2. What is the difference between equivalent figures and congruent figures?

3. State two propositions relating to the ratio of one rectangle to another.

4. Given the base and altitude of a rectangle, how is the area found? Given the area and base, how is the altitude found?

5. How do you justify the expression, "the product of two lines"? "the quotient of an area by a line”?

6. Can a triangle with a perimeter of 10 in. have the same area as one with a perimeter of 1 in.? Is the same answer true for two squares?

7. Can a parallelogram with a perimeter of 10 in. have the same area as a rectangle with a perimeter of 1 in.? Is the same answer true for two rectangles ?

8. Explain how the area of an irregular field with straight sides may be found by the use of the theorems of Book IV.

9. A triangle has two sides 5 and 6, including an angle of 70°, and another triangle has two sides 2 and 71⁄2, including an angle of 70°. What is the ratio of the areas of the triangles ?

10. Two similar triangles have two corresponding sides 5 in. and 15 in. respectively. The larger triangle has how many times the area of the smaller?

11. Given the hypotenuse of an isosceles right triangle, how do you proceed to find the area?

12. Given three sides of a triangle, what test can you apply to determine whether or not it is a right triangle?

13. Suppose you wish to construct a square equivalent to a given polygon, how do you proceed?

BOOK V

REGULAR POLYGONS AND CIRCLES

357. Regular Polygon. A polygon that is both equiangular and equilateral is called a regular polygon.

Familiar examples of regular polygons are the equilateral triangle and the square.

It is proved in Prop. I (§ 362) that a circle may be circumscribed about, and a circle may be inscribed in, any regular polygon, and that these circles are concentric (§ 188).

358. Radius. The radius of the circle circumscribed about a regular polygon is called the radius of the polygon.

In this figure r is the radius of the polygon.

α

m

359. Apothem. The radius of the circle inscribed in a regular polygon is called the apothem of the polygon.

In the figure a is the apothem of the polygon. The apothem is evidently perpendicular to the side of the regular polygon (§ 185).

360. Center. The common center of the circles circumscribed about and inscribed in a regular polygon is called the center of the polygon.

In the figure O is the center of the polygon.

361. Angle at the Center. The angle between the radii drawn to the extremities of any side of a regular polygon is called the angle at the center of the polygon.

In the figure m is the angle at the center of the polygon. It is evidently subtended by the chord which is the side of the inscribed polygon.