CHAPTER XXIV (Supplementary) MENSURATION OF THE TRIANGLE The basic method is: The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. (Art. 282.) 338. What is the Projection of a point upon a line? A line upon a line? 339. The square of the side opposite an acute angle of a triangle equals the sum of the squares of the other two sides minus twice the product of one of those sides and the projection of the other upon it. 340. The square of the side opposite an obtuse angle of a triangle equals the sum of the squares of the other two sides plus twice the product of one of those sides and the projection of the other upon it. 341. Group articles 282, 339, and 340 under one general statement. 342. The sum of the squares of two sides of a triangle equals twice the square of half the third side plus twice the square of the median to that side. 343. The product of two sides of a triangle equals the diameter of the circumscribed circle multiplied by the altitude to the third side. 344. The product of two sides of a triangle equals the square of the bisector of the included angle plus the product of the parts of the third side made by the bisector. 345. If s is the semi-perimeter of a triangle with sides a, b, and c, its area = √ s (s − a) (s – b) (s − c). 346. If AB is a side of a regular inscribed polygon with radius R, the side of a regular inscribed polygon of double the number of sides = √2R2 - R√ 4R2-AB2. 2R2 4R2 347. PROBLEM: Compute the value of π. SUMMARY 348. If the three sides of a triangle are known, state the formula for finding: R a. Whether the angles are acute, right, or obtuse. b. The projection of one side upon another. c. An altitude. d. The medians. e. The diameter of the circumscribed circle. f. The angle-bisectors. g. The area. 349. EXERCISES 1. If the radius of a circle is 1 in., show that: In a regular inscribed A, (a side) a = √3; (apothem) ↑ = 12; ZA = 60°. 2. Also, in a regular inscribed square, α = √2; r = 90°; ZBOC LA = = 90°. = 3. Also, in a regular inscribed hexagon, a = 1; r 120°; ZBOC: = 60°. 4. Also, in a regular circumscribed ▲, a = 2√3; (radius of ▲) 2. - = = 2; R = √2. 5. Also, in a regular circumscribed square, a = 6. Also, in a regular circumscribed hexagon, a = 23√3; R = 23√3. 7. If the sides of a triangle are 3, 4, and 5, is the angle opposite 5 acute, right, or obtuse? 8. If the sides of a triangle are 7, 8, and 12, is the angle opposite 12 acute, right, or obtuse? Opposite 7? 9. If the sides of a triangle are 6, 9, and 12, find the altitudes; the medians; the angle-bisectors; the diameter of the circumscribed circle. MENSURATION OF THE TRIANGLE 75 10. If the sides of a triangle are 10, 14, and 16, find the median and the altitude to 14. 11. If the sides of a triangle are 4, 5, and 6, find the bisector of the angle opposite 6 and the altitude to 6. 12. Find the area of the triangle in ex. 7; ex. 8; ex. 9; ex. 10; ex. 11. 13. Find the diameter of the circumscribed circle in ex. 7, ex. 8, ex. 9, ex. 10, ex. 11. 14. If AB is a side of a regular inscribed polygon with radius 1, derive the formula for the side of a regular inscribed polygon of double the number of sides. 15. If the side of a regular inscribed hexagon is 1, find the side of a regular inscribed dodecagon. 16. The sum of the squares of the sides of a parallelogram equals the sum of the squares of the diagonals. 17. If the sum of the squares of two sides of a triangle equals the square of the third side, the triangle is a right triangle. REVIEW QUESTIONS 1. State the theorem of limits. 2. In what kind of figure is it used? 3. What does represent? What is its commonly accepted value? 4. State the formula for the circumference in terms of π. 5. How can a line of length √3 be constructed? 6. In similar triangles, are any two corresponding lines in proportion with any other two corresponding lines? 7. How is the area of a sector found? 8. How is the area of a segment of a circle found? 9. State the formula for the diagonal of a square in terms of its side. 10. State the formula for the altitude of an equilateral triangle. 11. State the formula for the area of an equilateral triangle. CHAPTER XXV (Supplementary) 353. sides). FURTHER RATIOS 350. Define Extreme and Mean Ratio. 351. PROBLEM: Divide a line in extreme and mean ratio. 352. PROBLEM: Construct an angle of 36°. PROBLEM: Inscribe in a circle a regular decagon (10 354. PROBLEM: Construct a square having a given ratio to a given square. 355. PROBLEM: Construct a polygon similar to a given polygon and having a given ratio to it. 356. PROBLEM: Construct a polygon similar to a given polygon and equal to a second given polygon. 357. What are incommensurable quantities? (Art. 202.) State the theorem of limits. (Art. 293.) 358. In the same or equal circles two central angles have the same ratio as their intercepted arcs. 359. Three or more parallel lines intercept proportional parts on two transversals. (The incommensurable case of Art. 237.) 360. The area of a rectangle equals the product of its base and altitude. (The incommensurable case of Art. 300.) 361. EXERCISES 1. Divide internally in extreme and mean ratio a line 20 meters long; externally. 2. Divide internally in extreme and mean ratio a line 4 ft. long; externally. 3. Inscribe a regular pentagon (5 sides) in a circle. 5. 4. Inscribe a regular pentadecagon (15 sides) in a circle. (Construct central angles of 24°. That is, 60° - 36°.) Construct a square 6. Construct a square 7. Construct a circle 8. as large as a given square. as large as a given square. as large as a given circle. Construct a circle five times as large as a given circle. MISCELLANEOUS EXERCISES 1. Find the area of a triangle whose sides are 13, 14, and 15. 2. What is the area of a parallelogram whose base is 9 in. and altitude 4 in.? What is the area of a triangle having the same base and altitude? 3. The base of a triangle is 12 and its altitude is 9. The bases of an equal trapezoid are 13 and 14; what is its altitude? 4. The base of a triangle is 8. Find the base of a similar triangle 5 times as large; twice as large. 5. If similar polygons are constructed upon the three sides of a right triangle, the polygon on the hypotenuse equals the sum of the other two. 6. If through the middle point of one of the non-parallel sides of a trapezoid a line is drawn parallel to the opposite side, the parallelogram formed is equal to the trapezoid. 7. If two equal triangles have the same base and lie on opposite sides of it, the line ajoining their vertices is bisected by the base. 8. Two triangles having an angle of the one supplementary to an angle of the other are to each other as the products of the sides including the supplementary angles. 9. Show by drawing a geometric figure that: (a + b)2 = a2 + 2 ab + b2. 10. (a - b)2= a2 - 2ab + b2. 14. Find the area of a triangle whose sides are 7, 10, and 13. 15. Find the area of a triangle whose sides are 17, 25, and 28. 16. Find the three altitudes of a triangle whose sides are 7, 10, and 11. 17. Find the three altitudes of a triangle whose side are 8, 12, and 16. |