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, having the sum of its base and altitude equal to a given line.

K

R

HE

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.

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. Construct a square that shall be four fifths of a given triangle.

Ex. 1275. 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.]

388. DEF. A regular polygon is a polygon which is both equiangular and equilateral.

389. A circle can be circumscribed about any regular polygon.

Given ABCDE, a regular polygon of n sides.

Το

prove that a circle can be circumscribed about ABCDE. Proof. Construct a circle through A, B, and C, and let o be its center.

Draw OA, OB, OC, and OD.

LABC BCD.

(Why?)

LOBC=OCB.

(Why?)

·. L OBA = ≤ OCD.

In like manner, it may be proved that the circle passes through the remaining vertices of the polygon.

.. a circle can be circumscribed about the given polygon.

PROPOSITION II. THEOREM

Q. E. D.

390. A circle can be inscribed in any regular polygon.

HINT. Circumscribe a circle about the given polygon and prove that

the center is equidistant from the sides.

391. DEF. The center (0) of a regular polygon is the common center of the circumscribed and inscribed circles of the polygon.

392. DEF. The radius of a E regular polygon is the radius of the circumscribed circle; as OA.

393. DEF. The central angle is the angle between two radii drawn to the ends of one side; as AOB.

B

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