Ex. 389. To construct a triangle equivalent to the sum of two given triangles. AALA a BA αν с 2b Let ABC and A'B'C' be the given A, and a and a' their bases. To construct a ▲ equivalent to the sum of the ▲ ABC and A'B'C'. Construction. Draw h and h', the altitudes of the AABC and A'B'C'. Find c, the mean proportional between a and h, and c', the mean proportional between a' and h'. Find b so that b2 = c2 + c'2. § 388 § 417 Construct the A DEF with base equal to 2b, and altitude equal to b. Then the ADEF is the required. Proof. But and The area of the ▲ DEF = (2b × b) == = b2 = c2 + c′2 = † (a × h) + } (a′ × h'). the area of the ABC = the area of the AA'B'C' = (a × h), (a' × h'). .. Δ DEF ~ Δ ΑΒΓ + Δ Α' Β' Γ'. § 403 Const. § 403 Q. E. F. Ex. 390. To construct a triangle equivalent to the difference of two given triangles. Let ABC and A'B'C' be the given A, and a and a' their bases. ABC and A'B'C'. To construct a ▲ equivalent to the difference of the AA'B'C' and ABC. Construction. Draw h and h', the altitudes of the Find c, the mean proportional between a and proportional between a' and h'. Find b so that b2 c'2 — c2. = h, and c', the mean § 388 § 418 Construct the ▲ DEF with base equal to 2b, and altitude equal to b. Then the DEF is the ▲ required. Proof. The area of the ▲ DEF = (2 b × b) Ex. 391. To transform a given triangle into an equivalent equilateral triangle. Find EF, the mean proportional between AD and AB. On EF construct the equilateral ▲ EFG. § 388 § 312 § 136 § 410 Const. Ax. 1 $ 404 Ax. 1 Q. E. F. Ex. 392. To transform a parallelogram into an equivalent parallelogram having one side equal to a given length. Let ABCD be the given, a its base, and a' the given length. Find h', the fourth proportional to a', a, and h. Construct the A'B'C'D' on a' as base, with h' as altitude. a' § 386 Ex. 393. To transform a parallelogram into an equivalent parallelogram having one angle equal to a given angle. Ex. 394. To transform a parallelogram into an equivalent rectangle having a given altitude. Let ABCD be the given, a its base, and h' the given altitude. Then the rectangle A'B'C'D' is the rectangle required. Ex. 395. To transform a square into an equivalent equilateral triangle. Let ABCD be the given square. To transform the square ABCD into an equivalent equilateral ▲. Construction. Prolong BC to E, making E Then the AGHM is the ▲ required. Ex. 396. To transform a square into an equivalent right triangle having one leg equal to a given length. Let ABCD be the given square, and EF the given length. To transform the square ABCD into an equivalent rt. ▲ having one leg equal to EF. Construction. Find EG, the fourth pro portional to EF, 2 AB, and AB. § 386 D Construct the rt. A EFG with legs equal to EF and EG. Then the A EFG is the Proof. But and Ex. 397. required. (EF × EG) = the area of the ▲ EFG, AB2 ..A EFG the area of the square ABCD. the square ABCD. To transform a square into an equivalent rectangle having one side equal to a given length. Let ABCD be the given square, and EF the given length. D с H G To transform the square ABCD into an equivalent rectangle having one side equal to EF. Construction. Find EH, the third proportional to EF and AB. .. the rectangle EFGH≈ the square ABCD. Ex. 398. To construct a square equivalent to five eighths of a given $ 398 $ 398 Find GH, the mean proportional between AB and BE. Proof. But .. the rectangle ABEF≈ the square GHOP. § 388 Const. § 327 § 398 § 396 Q. E. F. Ex. 399. To construct a square equivalent to three fifths of a given pentagon. Let P represent the given pentagon. To construct a square equivalent to three fifths of P. H G Find EF, the mean proportional between § AB and † CD. § 388 Ex. 400. To divide a given triangle into two equivalent parts by a line through a given point P in one of the sides. Let ABC be the given A, and P the given point. To draw a line through P so as to divide the ▲ ABC into two equivalent parts. Construction. Draw CD to the middle point of AB, and draw PD. Draw CE to PD, and draw PE. A B DE Then PE is the line required. |