2. The area of an equilateral triangle circumscribed about a circle is four times the area of an inscribed equilateral triangle. B D E 3. The bisector of an angle of a triangle intersects the opposite side in D and the circumscribed circle in E. Prove that AB × AC = AD × AE. Draw EC, and show that ▲ ABD ~ ▲ AEC. 6. The median drawn to any side of a triangle bisects any line segment parallel to that side and included between the other two sides. 7. Two triangles are similar if their sides are respectively perpendicular. SUGGESTION: Show that A'B'C' and B are each supplementary to Z DB'C'. A' B C' 8. The median drawn to the bases of a trapezoid (not a parallelogram) and the other two sides meet in a point if sufficiently produced. 9. If a point P is joined to the vertices of a triangle ABC, through any point E on PA a line parallel to AB is drawn, meeting PB in F; through F a line parallel to BC, meeting PC in G; and G is joined to E; the triangle EFG is similar to the triangle ABC. 419. Indirect measurement by proportion. EXAMPLE: In the absence of more accurate surveying instru ments, the height of an object, as a tree, may be estimated by the following method: Let the observer hold a staff in the vertical position AB, and sight along the line OD to the top of the tree, having an assistant mark the point B on the staff; then sight along the horizontal line OC, while the assistant marks the points A, C on the staff D B m n A C E d FIG. 190. and tree, respectively. Now measure m, n, and d (= OC). The height x above the horizontal line may then be computed as follows: From similar ▲ OAB and OCD, REMARK. It is important to observe that m and n need not be measured in terms of the same unit as d. All that is essential is that the number corresponding to the ratio be determined. m The principle involved in this example is the basis of all practical methods of indirect measurement. In more refined methods, instead of using a vertical staff AB to determine the ratio a transit (see p. 176), a sextant, or a similar instrument m n is used. By means of these instruments the ≤ COD is measured directly, and the ratiom, corresponding to this particular angle, is n taken from tables in books specially prepared for this purpose. Thus if COD = 45°, it can be easily shown that m = 1. n 420. Trigonometric ratios. Functions of an acute angle. Let XAY be an acute angle. From points C1, C2 on AX draw perpendiculars to AX, meeting AY in B1, B2, respectively. That is, the ratios (1), (2), (3) are constant (do not vary for different positions of C) for any particular angle XAY. These ratios are called functions of the acute angle A, and are distinguished by the names sine, cosine, and tangent of angle A, respectively. Thus, referring to Fig. 192, in which the sides of the right triangle ACB are replaced by the corresponding small letters a, c, b; 421. Functions of 45°, 60°, and 30°. In order that the student may clearly comprehend that the trigonometric ratios really have numerical values for any particular angle, we shall compute a table of functions for angles of 45°, 60°, and 30°. 1. Let XAY be an angle of 45°. From a point C on AX draw a perpendicular to AX, meeting AY in B. Represent AC by n. Y -X A n C FIG. 193. Y B 2. Let XAY be an angle of 60°, etc. Then Z ABC 30°. Why? = Represent AC by n. Then AB=2n. Why? .. CB2 = (2 n)2 — n2 = 3 n2. Why? That is, CB= n√3. 1. To find the functions of 45° by construction and measurement. Construct an angle XAY = 45°. Lay off AB = 1 in., and draw CBLAX. Measure CB and AC to one hundredth of an inch. (Use a diagonal scale, 2. To find the functions of 60° by construction and measurement. 3. To find the functions of 30° by construction and measurement. 423. The transit. The figure shows a transit commonly used by surveyors for measuring angles. The instrument consists of H FIG. 195. B F the following essential parts: AB is a telescope giving an accurate line of vision, intersecting a horizontal axis EF, and a vertical axis GH, at right angles. The telescope AB may be turned about the axis EF, generating vertical angles, which may be read on a graduated circle C; it may also be turned about the axis GH, generating horizontal angles, which may be read on a graduated circle D. The vertical axis GH can be made plumb by suitable attachments. *For a more complete table of functions see page 218. |