PROP.XIII.—Theorem.
PROP.XIII.—Theorem.
In any triangle(ABC), the square on any side subtending an acute angle(C)is lessthan the sum of the squares on the sides containing that angle, by twice the rectangle(BC,CD)contained by either of them(BC)and the intercept(CD)betweenthe acute angle and the foot of the perpendicular on it from the oppositeangle.
Dem.—BecauseBCis divided into two segments inD,
BC2 + CD2 = BD2 + 2BC.CD [vii.];
Hence, adding, since
ThereforeAB2is less thanBC2+AC2by2BC.CD.
Or thus:Describe squares on the sides. DrawAE,BF,CGperpendicular to the sides; then, as in the demonstration of [I.xlvii.], the rectangleBGis equal toBE;AGtoAF, andCEtoCF.Hence the sum of the squares onAC,CBexceeds thesquare onABby twiceCE—that is, by2BC.CD.
Observation.—By comparing the proofs of the pairs of Props.iv.andvii.;v.andvi.;ix.andx.;xii.andxiii., it will be seen that they are virtually identical. In order to render this identitymore apparent, we have made some slight alterations in the usual proofs. The pairs of Propositionsthus grouped are considered in Modern Geometry not as distinct, but each pair is regarded as oneProposition.
Exercises.
Exercises.
1.If the angleCof the△ACBbe equal to an angle of an equilateral△,AB2=AC2+BC2−AC.BC.
2.The sum of the squares on the diagonals of a quadrilateral, together with four times thesquare on the line joining their middle points, is equal to the sum of the squares on itssides.
3.Find a pointCin a given lineABproduced, so thatAC2+BC2= 2AC.BC.
PROP.XIV.—Problem.To construct a square equal to a given rectilineal figure(X).
PROP.XIV.—Problem.To construct a square equal to a given rectilineal figure(X).
Sol.—Construct [I.xlv.] the rectangleACequal toX. Then, if the adjacent sidesAB,BCbe equal,ACis a square, and the problem is solved; if not, produceABtoE, and makeBEequal toBC; bisectAEinF; withFas centre andFEas radius, describe the semicircleAGE; produceCBto meet it inG.The squaredescribed onBGwill be equal toX.
Dem.—JoinFG. Then becauseAEis divided equally inFand unequally inB, the rectangleAB.BE, together withFB2is equal toFE2[v.], that is, toFG2; butFG2is equal toFB2+BG2[I.xlvii.]. Therefore the rectangleAB.BE+FB2is equal toFB2+BG2. RejectFB2, which is common, and we have the rectangleAB.BE=BG2; but sinceBEis equal toBC, the rectangleAB.BEis equal to the figureAC. ThereforeBG2is equal to the figureAC,and therefore equal to the givenrectilineal figure(X).
Cor.—The square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter.
Exercises.
Exercises.
1.Given the difference of the squares on two lines and their rectangle; find the lines.
2.Divide a given line, so that the rectangle contained by another given line and one segmentmay be equal to the square on the other segment.
Questions for Examination on Book II.
Questions for Examination on Book II.
1.What is the subject-matter of Book II.?Ans.Theory of rectangles.
2.What is a rectangle? A gnomon?
3.What is a square inch? A square foot? A square perch? A square mile?Ans.The squaredescribed on a line whose length is an inch, a foot, a perch, &c.
4.What is the difference between linear and superficial measurement?Ans.Linearmeasurement has but one dimension; superficial has two.
5.When is a line said to be divided internally? When externally?
6.How is the area of a rectangle found?
7.How is a line divided so that the rectangle contained by its segments may be amaximum?
8.How is the area of a parallelogram found?
9.What is the altitude of a parallelogram whose base is 65 metres and area 1430 squaremetres?
10.How is a line divided when the sum of the squares on its segments is a minimum?
11.The area of a rectangle is 108.60 square metres and its perimeter is 48.20 linear metres; findits dimensions.
12.What Proposition in Book II.expresses the distributive law of multiplication?
13.On what proposition is the rule for extracting the square root founded?
14.Compare I.xlvii.and II.xii.andxiii.
15.If the sides of a triangle be expressed byx2+ 1,x2−1, and 2xlinear units, respectively;prove that it is right-angled.
16.How would you construct a square whose area would be exactly an acre? Give a solution byI.xlvii.
17.What is meant by incommensurable lines? Give an example from Book II.
18.Prove that a side and the diagonal of a square are incommensurable.
19.The diagonals of a lozenge are 16 and 30 metres respectively; find the length of aside.
20.The diagonal of a rectangle is 4.25 perches, and its area is 7.50 square perches; what are itsdimensions?
21.The three sides of a triangle are 8, 11, 15; prove that it has an obtuse angle.
22.The sides of a triangle are 13, 14, 15; find the lengths of its medians; also the lengths of itsperpendiculars, and prove that all its angles are acute.
23.If the sides of a triangle be expressed bym2+n2,m2−n2, and 2mnlinear units,respectively; prove that it is right-angled.
24.If on each side of a square containing 5.29 square perches we measure from the cornersrespectively a distance of 1.5 linear perches; find the area of the square formed by joining the pointsthus found.
Exercises on Book II.
Exercises on Book II.
1.The squares on the diagonals of a quadrilateral are together double the sum of the squares onthe lines joining the middle points of opposite sides.
2.If the medians of a triangle intersect inO,AB2+BC2+CA2= 3(OA2+OB2+OC2).
3.Through a given pointOdraw three linesOA,OB,OCof given lengths, such that theirextremities may be collinear, and thatAB=BC.
4.If in any quadrilateral two opposite sides be bisected, the sum of the squares on the other twosides, together with the sum of the squares on the diagonals, is equal to the sum of the squares onthe bisected sides, together with four times the square on the line joining the points ofbisection.
5.If squares be described on the sides of any triangle, the sum of the squares on the linesjoining the adjacent corners is equal to three times the sum of the squares on the sides of thetriangle.
6.Divide a given line into two parts, so that the rectangle contained by the whole and onesegment may be equal to any multiple of the square on the other segment.
7.IfPbe any point in the diameterABof a semicircle, andCDany parallel chord,then
2 2 2 2 CP + PD =AP + PB .
8.IfA,B,C,Dbe four collinear points taken in order,
AB.CD +BC.AD = AC.BD.
9.Three times the sum of the squares on the sides of any pentagon exceeds the sum of thesquares on its diagonals, by four times the sum of the squares on the lines joining the middle pointsof the diagonals.
10.In any triangle, three times the sum of the squares on the sides is equal to four times thesum of the squares on the medians.
11.If perpendiculars be drawn from the angular points of a square to any line, the sum of thesquares on the perpendiculars from one pair of opposite angles exceeds twice the rectangle of theperpendiculars from the other pair by the area of the square.
12.If the baseABof a triangle be divided inD, so thatmAD=nBD, then
2 2 2 2 2 mAC +nBC = mAD + nDB + (m+ n)CD .
13.If the pointDbe taken inABproduced, so thatmAD=nDB, then
mAC2 − nBC2 = mAD2 − nDB2+ (m− n)CD2.
14.Given the base of a triangle in magnitude and position, and the sum or the difference ofmtimes the square on one side andntimes the square on the other side, in magnitude, the locus of thevertex is a circle.
15.Any rectangle is equal to half the rectangle contained by the diagonals of squares describedon its adjacent sides.
16.IfA,B,C. &c., be any number of fixed points, andPa variable point, find the locus ofP, ifAP2+BP2+CP2+ &c., be given in magnitude.
17.If the area of a rectangle be given, its perimeter is a minimum when it is a square.
18.If a transversal cut in the pointsA,C,Bthree lines issuing from a pointD, provethat
BC.AD2 + AC.BD2 − AB.CD2 = AB.BC.CA.
19.Upon the segmentsAC,CBof a lineABequilateral triangles are described: prove that ifD,D′be the centres of circles described about these triangles, 6DD′2=AB2+AC2+CB2.
20.Ifa,b,pdenote the sides of a right-angled triangle about the right angle, and theperpendicular from the right angle on the hypotenuse,-12 a+12- b=12- p.
21.If, upon the greater segmentABof a lineAC, divided in extreme and mean ratio, anequilateral triangleABDbe described, andCDjoined,CD2= 2AB2.
22.If a variable line, whose extremities rest on the circumferences of two given concentriccircles, subtend a right angle at any fixed point, the locus of its middle point is a circle.