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PROPOSITION XI. PROBLEM.

381. To construct a square equivalent to the of two given squares.

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Take AC equal to a side of R',

AB equal to a side of R; and draw BC.

Construct the square S, having each of its sides equal S is the square required.

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(the square on the hypotenuse of a rt. ▲ is equivalent to the su squares on the two sides). ..SR'+R.

Ex. 296. If the perimeter of a rectangle is 72 feet, and the 1 equal to twice the width, find the area.

Ex. 297. How many tiles 9 inches long and 4 inches wide required to pave a path 8 feet wide surrounding a rectangular c feet long and 36 feet wide?

Ex. 298. The bases of a trapezoid are 16 feet and 10 feet;

382. To construct a square equivalent to the difference of two given squares.

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Let R be the smaller square and R' the larger.

To construct a square equivalent to R' — R.

Construction.

Construct the rt. Z A.

Take AB equal to a side of R.

From B as a centre, with a radius equal to a side of R', describe an arc cutting the line AX at C.

Construct the square S, having each of its sides equal to AC. S is the square required.

Proof.

2

AC BC-AB3,

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(the square on either leg of a rt. ▲ is equivalent to the difference of the squares on the hypotenuse and the other leg).

.. SR' R.

Q. E. F.

Ex. 299. Construct a square equivalent to the sum of two squares whose sides are 3 inches and 4 inches.

Ex. 300. Construct a square equivalent to the difference of two squares whose sides are 24 inches and 2 inches.

Ex. 301. Find the side of a square equivalent to the sum of two squares whose sides are 24 feet and 32 feet.

Ex. 302. Find the side of a square equivalent to the difference of two squares whose sides are 24 feet and 40 feet.

Ex. 303. A rhombus contains 100 square feet, and the length of one

dieconel is 10 foot Find the length of the other diegenel

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383. To construct a square equivalent to the of any number of given squares.

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Let m, n, o, p, r be sides of the given squares.

To construct a square

Construction.

The

m2 + n2 + o2 + p2 + r2.

Take AB = m.

Draw AC =n and I to AB at A, and draw B

Draw CE = 0 and 1 to
Draw EF=p and 1 to

BC at C, and draw E

BE at E, and draw E

Draw FH- r and 1 to BF at F, and draw E square constructed on BH is the square requir

Proof. BH'FH2 + BF2,

≈ FH2 + EF2 + EB2,

≈ FH2+ EF2+ EC2 + CB3,

≈ FH2 + EC2 + EF2 + CA2 + AB2,

(the sum of the squares on the two legs of a rt. A is equivalent to t

That is,

on the hypotenuse).

BH2 m2 + n2 + o2 + p2 + r22,

384. To construct a polygon similar to two given similar polygons and equivalent to their sum.

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Let R and R' be two similar polygons, and AB and A'B' two homologous sides.

To construct a similar polygon equivalent to R + R'.

Construction. Construct the rt. Z P.

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Take PHA'B', and PO= AB.

Draw OH, and take A"B" = OH.

Upon A"B", homologous to AB, construct R" similar to R. Then R is the polygon required.

Proof. PO+PH' = OH', :. AB2 + A'B'2 = A"B"2.

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(similar polygons are to each other as the squares of their homologous sides).

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385. To construct a polygon similar to two similar polygons and equivalent to their diffe

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Let R and R' be two similar polygons, and A'B' two homologous sides.

To construct a similar polygon equivalent to R'R Construction. Construct the rt. Z P,

and take PO = AB.

From O as a centre, with a radius equal to A'l describe an arc cutting PX at H, and join O Take A"B" = PH, and on A"B", homologous to construct R" similar to R.

Then R" is the polygon required.

Proof. PH OH - OP', .. A"B"2 = A'B'2 —,

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(similar polygons are to each other as the squares of their homolog

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