THEOREM 9.

In any triangle, the square on the side opposite to an acute angle is equal to the sum of the squares on the sides containing that acute angle minus twice the rectangle contained by one of those sides and the projection on it of the other.

CN is the perpendicular from C upon AB (or AB produced), .. AN is the projection of AC upon AB.

Let BC= a, CA =b, AB = c, AN = p, CN=h.

Ex. 1117.

angle.

Write out the proof of 11. 9 for the case in which ▲ B is a right
What does the theorem become?

Ex. 1118. Verify the truth of IL 8, 9 by drawing and measurement.

Ex. 1119. What is the area of the rectangle referred to in the enunciation of 11. 8, 9 for the following cases:

Ex. 1120. By comparing the square on one side with the sum of the squares on the two other sides, determine whether triangles having the following sides are acute-, obtuse-, or right-angled (check by drawing) :—

(i) 3, 4, 6; (ii) 3, 4, 3; (iii) 2, 3, 5; (iv) 2, 3, 4; (v) 12, 13, 5.

Ex. 1121. Given four sticks of lengths 2, 3, 4, 5 feet, how many triangles can be made by using three sticks at a time? Find out whether each triangle is acute-, obtuse-, or right-angled.

Ex. 1122. Calculate BC when

AB=10 cm., AC=8 cm., 4A=60°. (See Ex. 1113.)

Ex. 1123.

Calculate BC when

AB=10 cm., AC=8 cm., ▲ A=120°.

Ex. 1124. Bristol is 26 miles E. of Cardiff; Reading is 70 miles E. of Bristol; Naseby is due N. of Reading and 95 miles from Bristol. Calculate the distance from Cardiff to Naseby, and check by measurement

Ex. 1125. Brighton is 48 miles S. of London; Hertford is 20 miles N. of London; Shoeburyness is due E. of London, and 64 miles f Brighton. How far is it from Hertford? Verify graphically.

Revise Ex. 256.

Ex. 1126. Suppose that LA in fig. 208 becomes larger and larger til BAC is a straight line. What does 11. 8 become in this case?

Ex. 1127.

C is on BA.

Suppose that LA in fig. 209 becomes smaller and smaller till What does II. 9 become in this case?

Ex. 1128. In the trapezium ABCD (fig. 211), prove

that AC2+ BD2=AD2 + BC2 + 2AB. CD.

(Apply 11. 9 to ▲ ACD and BCD.)

Ex. 1129.

D is a point on the base BC of an isosceles ▲ ABC. Prove that AB2=AD2 + BD. CD.

(Let O be mid-point of BC, and suppose that D lies between B and O. Then

BD=BO-OD, CD=CO+ OD=BO + OD.)

Ex. 1130.

ABC is an isosceles ▲ (AB=AC); BN is an altitude. Prove that 2AC. CN=BC2.

Ex. 1131. BE, CF are altitudes of an acute-angled ▲ ABC. Prove that AE. AC AF. AB.

(Write down two different expressions for BC2.)

Ex. 1182. In the figure of Ex. 1131, BC2=AB. FB+AC. EC.

Ex. 1133. The sum of the squares on the two sides of a triangle ABC is equal to twice the sum of the squares on the median AD, and half the base. (Apollonius' theorem.)

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(Draw AN i to BC; apply 11. 8, 9 to ▲ ABD, ACD.)

Ex. 1134. Use Apollonius' theorem to calculate the lengths of the three medians in a triangle whose sides are 4, 6, 7. Check by drawing.

Ex. 1135. Repeat Ex. 1134, with sides 4, 5, 7.

s fi

jer til

Ex. 1136.

Calculate the base of a triangle whose sides are 8 cm. and

16 cm., and whose median is 12 cm. Verify graphically.

Revise Ex. 246.

Ex. 1137. The base BC of an isosceles A ABC is produced to D, so that CD=BC; prove that AD2=AC2+2BC2.

Ex. 1138. A side PR of an isosceles ▲ PQR is produced to S so that RS PR: prove that QS2-2QR2+ PR2.

Ex. 1139. The base AD of a triangle OAD is trisected in B, C. Prove that OA2+20D2=30C2+6CD2.

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(Apply Apollonius' theorem to ▲ OAC, OBD; then eliminate OB2.)

Ex. 1140. In the figure of Ex. 1139, OA2+ OD2 = OB2 + OC2+4BC2.

Ex. 1141. A point moves so that the sum of the squares of its distances from two fixed points A, B remains constant; prove that its locus is a circle, having for centre the mid-point of AB.

Ex. 1142. The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals.

Ex. 1143. In any quadrilateral the sum of the squares on the four sides exceeds the sum of the squares on the diagonals by four times the square on the straight line joining the mid-points of the diagonals.

(Let E, F be the mid-points of AC, BD; apply Apollonius' theorem to A BAD, BCD and AFC.)

Ex. 1144. The sum of the squares on the diagonals of a quadrilateral is equal to twice the sum of the squares on the lines joining the mid-points of opposite sides. (See Ex. 736 and 1142.)

Ex. 1145. In a triangle, three times the sum of the squares on the sides =four times the sum of the squares on the medians.

Ex. 1146. What does Apollonius' theorem become if the vertex moves down (i) on to the base, (ii) on to the base produced?

1.

CAMBRIDGE: PRINTED BY J. & C. F. CLAY, AT THE UNIVERSITY PRESS.

DEF. A circle is a line, lying in a plane, such that all points in the line are equidistant from a certain fixed point, called the centre of the circle.

In view of what has been said already about loci we may give the following alternative definition of a circle:

:

DEF. A circle is the locus of points that lie at a fixed distance from a fixed point (the centre). The fixed distance is called the radius of the circle.

The word "circle" has been defined above to mean a certain kind of curved line. The term is, however, often used to indicate the part of the plane inside this line. If any doubt exists as to the meaning, the line is

called the circumference of the circle.

Two circles are said to be equal if they have equal radii.

If one of two equal circles is applied to the other so that the centres coincide, then the circumferences also will coincide.

G. S. II.

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