Ex. 1222. The areas of two similar triangles are to each other as the squares of any two homologous medians.

Ex. 1223. If the median AX, BY, and CZ of ▲ ABC, meet in O, then ▲ AOC is equivalent to ▲ BOC.

Ex. 1224. If on the line joining the mid-points of two sides of a triangle a parallelogram is constructed having two of its vertices in the base of the triangle, the parallelogram is equivalent to of the triangles.

Ex. 1225. The non-parallel sides of a trapezoid form with the diagonals two equivalent triangles.

Ex. 1226. If from the point of intersection of the medians of any triangle, lines are drawn to the three vertices, they form with the sides three equivalent triangles.

Ex. 1227. If in the triangle ABC, D and F are the mid-points of the sides AB and AC respectively, the triangles ADC and ABF are equivalent.

a2

Ex. 1228. The area of an equilateral triangle whose side equals a is √3.

Ex. 1229. The area of a rhombus is equal to half the product of the diagonals.

Ex. 1230. If on the line joining the mid-points of the two non-parallel sides of a trapezoid, a triangle be constructed whose vertex lies in the upper base of the trapezoid, the triangle is equivalent to the trapezoid.

Ex. 1231. If, on two sides of triangle ABC, parallelograms DB and BG are constructed, their sides DE and GH be produced to meet in F, and on AC a parallelogram be constructed, having AK equal and parallel to FB, then the parallelogram AL is equivalent to parallelogram AE plus parallelogram BG. (Pappus' Theorem.)

Ex. 1232. Find a similar proposition for triangles constructed on the three sides of a given triangle.

B

H

G

C

*Ex. 1233. A quadrilateral is equivalent to a triangle if its diagonals and the angle included between them are respectively equal to two sides and the included angle of the triangle.

* Ex. 1234. Two quadrilaterals are equivalent if the diagonals and the included angle of one are equal, respectively, to the diagonals and the included angle of the other. (Ex. 1233.)

* Ex. 1235. If two chords intersect within a circle at right angles, the sum of the squares upon their segments is equal to the square of the diameter.

Ex. 1238. Find the area of a trapezoid whose bases are 9 and 11 respectively, and whose altitude is 12 ft.

Ex. 1239. Find the area of a rhombus whose diagonals are 9 and 10 ft. respectively.

Ex. 1240. Find the area of quadrilateral ABCD if AB = 10, BC =24, CD = 30, AD = 28, diagonal AC = 26.

Ex. 1241. A side of equilateral triangle ABC is 8. Find the side of an equilateral triangle equivalent to three times triangle ABC.

Ex. 1242. The perimeter of a rectangle is 20 m., one side is 6 m. Find the area.

Ex. 1243. n sq. m.?

Ex. 1244.

What is the side of a square whose area is 900 sq. m.?

The area of a rhombus is equal to m, and one diagonal is

equal to d. Find the other diagonal.

Ex. 1245.

The area of a trapezoid is 400 sq. m., its altitude is 8 m. Find the length of the line joining the mid-points of the non-parallel sides.

Ex. 1246. The hypotenuse of a right triangle is 20, and the projection of one arm upon the hypotenuse is 4. What is its area?

Ex. 1247. A farmer wishes to determine the area of a pentagonal field. He measures the lines AB = 4 rods, BC = 13 rods, CD = 14 rods, DE = 5 rods, EA 12 rods, AC: 15 rods, and AD 13 rods. How many square rods does the field contain?

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Ex. 1248. The base and altitude of a triangle are 12 and 20 respectively. At a distance of 6 from the base, a parallel is drawn to the base. Find the areas of the two parts of the triangle.

Ex. 1249. Find the area of a rectangle having one side equal to 6 and a diagonal equal to 10.

Ex. 1250. Find the area of a polygon whose perimeter equals 20 ft., circumscribed about a circle whose radius is 3 ft.

Ex. 1251. Find the area of a triangle whose base is 10 inches and whose base angles are 120° and 30° respectively.

Ex. 1252. Find the side of an equilateral triangle equivalent to a parallelogram, whose base and altitude are 10 and 15 respectively.

Ex. 1253. The chord of an arc is 42 in.; the chord of one half that arc is 29 in. Find the diameter of the circle.

Ex. 1254.

The base of a triangle is 15 ft., its area is 60 sq. ft.; find the area of a similar triangle whose altitude is 6 ft.

Ex. 1255. An arm of an isosceles trapezoid is 13 in. and its projection on the longer base is 5 in.; the longer base is 17 in. Find the area of the trapezoid.

Ex. 1256. The arms of two isosceles right triangles are 3 and 2 respectively. Find the arm of an isosceles right triangle equivalent to their difference.

Ex. 1257. The sides of a triangle are as 8:15: 17. Find the sides if the area is 960 sq. ft.

Ex. 1258. The sides of a triangle are 8, 15, and 17. Find the radius of the inscribed circle.

HINT. Express in the form of an equation the fact that the area of the triangle is equal to the sum of the three triangles whose vertex is the incenter and whose bases are the sides of the triangle.

Ex. 1259.

Find the area of an equilateral polygon of 12 sides inscribed in a circle whose radius is 4 in.

Ex. 1260. The sides of a triangle are 6, 7, and 8 ft. Find the areas of the two parts into which the triangle is divided by the bisector of the angle included by 6 and 7.

Ex. 1261. Find the area of an equilateral triangle whose altitude is equal to h.

PROBLEMS OF CONSTRUCTION

Ex. 1262. To construct a rectangle equivalent to a given square, ing the sum of its base and altitude equal to a given line.

hav

HINT. If FE is the given line, and ABCD the given square, make EK= AB.

Ex. 1263. To construct a rectangle equivalent to a given square, having the difference of its base and altitude equal to a given line.

G

R

HINT. If ABCD is the given square and EF the given line, make EG = AB.

Ex. 1264. To transform a rectangle into another one, having given one side.

Ex. 1265. To transform a square into an isosceles triangle, having a given base.

Ex. 1266. To transform a rectangle into a parallelogram, having a given diagonal.

Ex. 1267. To divide a triangle into three equivalent parts by lines parallel to a median.

Ex. 1268. To bisect a parallelogram by a line perpendicular to a side. Ex. 1269. To divide a parallelogram into three equivalent parts by lines drawn through a vertex.

Ex. 1270. To bisect a trapezoid by a line drawn through a vertex. Ex. 1271. Divide a pentagon into four equal parts by lines drawn through one of its vertices.

Ex. 1272. Divide a quadrilateral into four equal parts by lines drawn from a point in one of its sides. (Consider the figure a pentagon.)

Ex. 1273. Find a point within a triangle such that the lines joining the point to the vertices shall divide the triangle into three equivalent parts.

Ex. 1274.

triangle.

Ex. 1275.

Construct a square that shall be four fifths of a given

Construct an equilateral triangle that shall be two thirds of a given rectangle.

Ex. 1276. Find a point within a triangle such that the lines joining the point with the vertices shall form three triangles, having the ratio 3:4:5.

Ex. 1277. Divide a given line into two segments such that one segment is to the line as √2 is to √5.

Ex. 1278. To transform a triangle into a right isosceles triangle.

Ex. 1279. Construct a triangle similar to a given triangle and equivalent to another given triangle.

Ex. 1280. To divide a triangle into three equivalent parts by lines parallel to a given line.

*Ex. 1281. Bisect a trapezoid by a line parallel to the bases. [For practical applications see p. 300.]