261.

Cor. V. Can you show how to find the area of any triangle?

262.

Cor. VI. Can you show that triangles with equal bases and equal altitudes are equivalent?

263.

Cor. VII. Can you show that triangles with equal altitudes are to each other as their bases; and those with equal bases are to each other as their altitudes?

264.

Cor. VIII. Can you show that any two triangles are to each other as the products of their bases and altitudes.

EXERCISES.

205. The altitude and base of a being 35 and 12, respectively, what is its area?

206. The area of a ▲ is 221 square feet; its base is 5 yards. What is its altitude in inches?

207. The bases of two parallelograms are 15 cm. and 16 cm. respectively; and their altitudes are 8 cm. and 10 cm. respectively. What is the ratio of their areas?

208. Two As of equal areas have their bases 26 mm. and 36 mm. respectively. What is the ratio of their altitudes?

209. Draw any straight line through the point of intersection of the diagonals of a parallelogram terminating in a pair of opposite sides and show how the parallelogram is divided.

210. If E is the middle point of C D, one of the nonparallel sides of the trapezoid A B CD, prove that the triangle A B E [draw A E and B E] is equivalent to one-half the trapezoid.

211. If E and F are the middle points of the sides A B and A C of a triangle, and D is any point in B C, show how the quadrilateral A E D F is related to the ▲ A B C.

212. Join the middle points of the adjacent sides of any quadrilateral. What is the new figure? How is it related to the quadrilateral?

213. If two equivalent As have a common base and have their vertices on opposite sides of the base, the line joining their vertices is bisected by the base (produced if necessary).

214. Construct a parallelogram, A B C D, and draw the diagonal A C. Take any point, P, on A C and join it with B and D. Compare the areas of A B P and A D P, of B P C and D P C.

What is the figure A B C D if A B is D C? What is the altitude? What is its area?

[Hint.-What 2As compose ABCD? ABD=? BCD ? Therefore A B C D = ?]

Generalize this equation and call it Prop. VII.

266.

Cor. 1. Show that one-half the sum of the parallel bases equals the median of the trapezoid?

EXERCISES.

215. Altitude of a trapezoid is 5, bases 8 and 10, find area. 216. Construct an irregular pentagon, and, having compasses and rule, show how to compute area.

217. The area of a rhombus

its diagonals. Prove.

one-half the product of

218. Rough boards are usually narrower at one end than at the other, for which reason the lumber merchant usually measures their width in the middle. Can you explain the principle involved in such measurement?

219. A carpenter wishes a trapezoidal board whose nonparallel sides must be equal. He lays off equal angles with one of the bases and saws out his board.

[Let the sides of the s which coincide with the base of the trapezoid extend in opposite directions from the vertices.] Prove that his method is right.

220. Suppose the trapezoidal board mentioned in Ex. 219 simply required that the base angles made by the nonparallel sides should be equal. What could the carpenter discover concerning the non-parallel sides?

221. (1) Can you give two methods of finding the area of any polygon?

(2) Show that equiangular As are similar.

Given: The similar As A B C and A' B' C'.

To Show-That corresponding altitudes are to each other as any two homologous sides.

Sug. If CD and C' D' are homologous altitudes, can you show that A CD is similar to A' C' D'?

[When are triangles similar?]

=

=

Can you show that CD: C'D' AC: A'C' AB: A'B' = BC: B' C'?

Write the general truth as Prop. VIII.

Scholium: The ratio of any two homologous sides of similar polygons is called the ratio of similitude.

222. Can you show that any two similar ▲s are to each other as the squares of any two homologous lines, or are in the ratio of similitude of the triangles?

223. In Fig., § 267, if A B = 10, A' B'

A' B'C' 36, what is the area of A B C?

=

6 and area A

268.

PROPOSITION IX.

Fig.

Given: As A B C and A B' C'any 2 s having an angle of one equal to an angle of the other.

To Compare-A B C and A B'C'.

Sug. Compare A B C and A B' C. Compare A B'C and A B'C'. Express these two comparisons in fractional forms. Multiply the two equations together and simplify the result Express the result in a general statement. This is Prop. IX.

269.

Cor. I. If 2 parallelograms have an angle of the one equal to an angle in the other, how are they related?

224. Given the perimeter of a triangle and the radius of the inscribed circle to find its area.

270.

Cor. II. If the products of the sides about the equal angles are equal, what can you say of the triangles?