Ex. 321. A line is drawn parallel to a side AB of a triangle ABC, cutting AC in D, BC in E. If AD: DC2:3, and AB = 20 inches,

find DE.

Ex. 322. The sides of a triangle are 9, 12, 15. the sides made by bisecting the angles.

Find the segments of

Ex. 323. A tree casts a shadow 90 feet long, when a post 6 feet high casts a shadow 4 feet long. How high is the tree?

Ex. 324. The lower and upper bases of a trapezoid are a, b, respectively; and the altitude is h. Find the altitudes of the two triangles formed by producing the legs until they meet.

In a similar

Ex. 325. The sides of a triangle are 6, 7, 8, respectively. triangle the side homologous to 8 is 40. Find the other two sides.

Ex. 326. The perimeters of two similar polygons are 200 feet and 300 feet. If a side of the first is 24 feet, find the homologous side of the second.

Ex. 327. How long a ladder is required to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house?

Ex. 328. If the side of an equilateral triangle is a, find the altitude. Ex. 329. If the altitude of an equilateral triangle is h, find the side. Ex. 330. Find the length of the longest chord and of the shortest chord that can be drawn through a point 6 inches from the centre of a circle whose radius is 10 inches.

Ex. 331. The distance from the centre of a circle to a chord 10 feet long is 12 feet. Find the distance from the centre to a chord 24 feet long.

Ex. 332. The radius of a circle is 5 inches. Through a point 3 inches from the centre a diameter is drawn, and also a chord perpendicular to the diameter. Find the length of this chord, and the distance from one end of the chord to the ends of the diameter.

Ex. 333. The radius of a circle is 6 inches. Find the lengths of the tangents drawn from a point 10 inches from the centre, and also the length of the chord joining the points of contact.

Ex. 334. The sides of a triangle are 407 feet, 368 feet, and 351 feet. Find the three bisectors and the three altitudes.

し

Ex. 335. If a chord 8 inches long is 3 inches distant from the centre of the circle, find the radius, and the chords drawn from the end of the chord to the ends of the diameter which bisects the chord.

Ex. 336. From the end of a tangent 20 inches long a secant is drawn through the centre of the circle. If the external segment of this secant is

8 inches, find the radius of the circle.

Ex. 337. The radius of a circle is 13 inches. Through a point 5 inches from the centre any chord is drawn. What is the product of the two segments of the chord? What is the length of the shortest chord that can be drawn through the point?

Ex. 338. The radius of a circle is 9 inches and the length of a tangent 12 inches. Find the length of a line drawn from the extremity of the tangent to the centre of the circle.

Ex. 339. Two circles have radii of 8 inches and 3 inches, respectively, and the distance between their centres is 15 inches. Find the lengths of

their common tangents.

Ex. 340. Find the segments of a line 10 inches long divided in extreme and mean ratio.

Ex. 341. The sides of a triangle are 4, 5, 5. Is the largest angle acute, right, or obtuse?

Ex. 342. Find the third proportional to two lines whose lengths are 28 feet and 42 feet.

Ex. 343. If the sides of a triangle are a, b, c, respectively, find the lengths of the three altitudes.

Ex. 344. The diameter of a circle is 30 feet and is divided into five equal parts. Find the lengths of the chords drawn through the points of division perpendicular to the diameter.

Ex. 345. The radius of a circle is 2 inches. From a point 4 inches from the centre a secant is drawn so that the internal segment is 1 inch. Find the length of the secant.

Ex. 346. The sides of a triangular pasture are 1551 yards, 2068 yards, 2585 yards. Find the median to the longest side.

Ex. 347. The diagonal of a rectangle is d, and the perimeter is p. Find the sides.

Ex. 348. The radius of a circle is r. distance from the centre is r.

Find the length of a chord whose

BOOK IV.

AREAS OF POLYGONS.

392. DEF. The unit of surface is a square whose side is a unit of length.

393. DEF. The area of a surface is the number of units of surface it contains.

394. DEF. Plane figures that have equal areas but cannot be made to coincide are called equivalent.

"tri

NOTE. In propositions relating to areas, the words "rectangle," angle," etc., are often used for "area of rectangle, 9966 area of triangle," etc.

PROPOSITION I. THEOREM.

395. Two rectangles having equal altitudes are to each other as their bases.

Let the rectangles AC and AF have the same altitude AD.
To prove that rect. AC: rect. AF base AB: base AE.
CASE 1. When AB and AE are commensurable.

Proof. Suppose AB and AE have a common measure, as AO, which is contained m times in AB and n times in AE.

Apply 40 as a unit of measure to AB and AE, and at the several points of division erect Is.

Proof. Divide AB into any number of equal parts, and apply one of them to AE as many times as AE will contain it.

Since AB and AE are incommensurable, a certain number of these parts will extend from A to some point K, leaving a remainder KE less than one of the equal parts of AB.

If the number of equal parts into which AB is divided is indefinitely increased, the varying values of these ratios will continue equal, and approach for their respective limits the ratios

rect. AF

and rect. AC

AE
AB

(See § 287.)

396. COR. Two rectangles having equal bases are to each

other as their altitudes.

PROPOSITION II. THEOREM.

397. Two rectangles are to each other as the products of their bases by their altitudes.

Let R and R' be two rectangles, having for their bases b and b', and for their altitudes a and a', respectively.

Proof. Construct the rectangle S, with its base equal to that of R, and its altitude equal to that of R'.

The products of the corresponding members of these equations give

R

R

ахъ a' x b'

Q. E. D.

Ex. 349. Find the ratio of a rectangular lawn 72 yards by 49 yards to a grass turf 18 inches by 14 inches.

Ex. 350. Find the ratio of a rectangular courtyard 18 yards by 15 yards to a flagstone 31 inches by 18 inches.

Ex. 351. A square and a rectangle have the same perimeter, 100 yards. The length of the rectangle is 4 times its breadth. Compare their areas.

Ex. 352. On a certain map the linear scale is 1 inch to 5 miles. How many acres are represented on this map by a square the perimeter of which is 1 inch?