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16. In the previous question, if F be the middle point of BC, and OF be joined, what relation holds between the areas of the six triangles OAD, ODB, . . . with vertex at O ?
17. In the same question, what is the position of AO, OF with respect to each other? Test and give reasons.
18. Construct two equal triangles on the same base and on opposite sides of it. What is the only restriction as to the positions of their vertices? If the vertices be joined, how is the joining line divided by the base, or base produced ?
19. From any point in an equilateral triangle draw perpendiculars to the sides. What relation exists between their sum and the altitude of the triangle? Give reasons.
20. The sides of a right-angled triangle are 3, 4 and 5 inches. If a Point within the triangle be 1 inch from each of the sides containing the right angle, how far is it from the hypotenuse?
21. In a quadrilateral ABCD, AB=2, BC=3, and CD=1} inches. ABC=35°, BCD=100°. Construct a triangle and also a rectangle equal to it in area. Hence calculate the area of ABCD, approximately, in square inches.
Squares on Sides of a Right-angled Triangle.
1. Let the angle B of the triangle ABC be 90°. Describe squares on the sides of ABC, as in the figure. Draw the lines AG, EF, CH parallel to BC; and the lines DK, EH parallel to AB.
B Then measurement (with dividers for lines, and bevel for angles) will show that the triangle AGD is in all respects equal to the triangle ABC; and cutting out the triangle AGD, it may be turned about A, in the direction indicated by the arrow head, into the position ABC. Measurement will also show that the triangle EFD is in all respects equal to the triangle EHC; and cutting out the triangle EFD, it may be turned about E, in the direction indicated by the arrow head, into the position EHC. We thus have the square on AC converted into ABKG and FKHE, which will be found to be the squares on AB and BC.
Repeat the same construction, measurements, and superposition in the case of the following triangles:
AB=35, BC=50 millimetres; ABC=90°.
The result of these observations may be stated thus: In any right-angled triangle the square which is described on the side subtending the right angle, is equal to the sum of the squares described on the sides containing the right angle.
Two sides of a right-angled triangle about the right angle, are 3 and 4. What is the length of the third side ?
If a string or rope of length 12 be broken into lengths 3, 4 and 5, and these be formed into a triangle, such triangle is right-angled.
If the lengths of the pieces of rope be 30, 40 and 50, the triangle formed with them is also right-angled.
2. A square may be constructed equal in area to any rectangle, as follows:
Let ABCD be the rectangle. Make DE equal to DC, and find F the middle point of AE. Describe the semicircle, and produce CD to G. Then the square on DG is equal to the rectangle ABCD.
For, describe the square DGLK on DG, and let LK and BC meet in H. Then, if the figure has been accurately constructed, on producing the lines LG, HD and BA, they will be found to all pass through one point, M. Hence (Ch. VIII., 6) the square GDKL is equal to the rectangle ABCD,
In the succeeding constructions it is of course absolutely necessary that the three lines corresponding to LG, HD and BA pass through the same point (M).
Describe the rectangle whose sides are 40 and 90 millimetres. Construct, as above, the square equal to it. Measure in millimetres the side of the square, and thence verify the accuracy of your construction.
Proceed similarly with the rectangle whose sides are 1 and 4 inches.
Also with the rectangle whose sides are 9 and 16 sixteenths of an inch.
Also with the rectangle whose sides are 18 and 32 sixteenths of an inch. The sides of this rectangle are twice those of the former: note the number of times its area is greater than that of the former; note the same with respect to the resulting squares.
ABC is a right-angled triangle, ABC being the right angle; and BD is perpendicular to AC. Construct the rectangle whose
A4 sides are CA, AD; by the preceding method construct the square equal to it, and show that it is the square on AB.
Similarly by construction show that the rectangle contained by AC, CD is equal to the square on BC.
Also that the rectangle contained by AD, DC is equal to the square on BD.
Exercises. 1. Three straight lines, of lengths 3, 4, 5, forming a right-angled triangle, what sort of triangle is formed by lines of lengths 6, 8, 10, or 9, 12, 15, or 12, 16, 20, etc.?
2. Construct triangles with sides as follows : 3, 4, 5 inches ; 30, 40, 50 millimetres ; 36, 48, 60 millimetres ; 34, 5, 64 inches. Compare the angles of these triangles, and state the result of such comparison. What relation do the sides of one triangle bear to the sides of another ?
(2n+1) + (2n+2n) = (2n2 + 2n+1)? by assigning to n in succession the values 1, 2, 3, form a series of whole numbers, in groups of three, such that each group gives the lengths of the sides of a right-angled triangle.
4. The side of an equilateral triangle is 2. What is the length of the perpendicular from any angle on the opposite side ?
5. Draw two lines CA, CB at right angles to each other and each of length one inch. What is the area of the square on AB ?
6. In the figure of the preceding question, draw AD (=1 in.) perpendicular to AB. What is the area of the square on DB?
7. In the same figure draw DE (= 1 in.) perpendicular to DB. What is the area of the square on EB? Test by measuring the length of EB.
8. Construct a square which shall contain 13 square inches.
9. Test the accuracy of the construction in the preceding question by drawing, at right angles to the side of the square, a line equal to the side of a square containing 3 square inches, joining the ends of the lines, and measuring the hypotenuse of the right-angled triangle so obtained.
10. Describe squares on the sides of a right-angled triangle. Construct another triangle with sides equal to the diagonals of these squares.
What is this latter triangle? 11. In the preceding question by what multiplier can you obtain the sides of one triangle from those of the other ?
Compare the angles of the two triangles and state the result of such comparison.
12. Describe triangle such that the square on one side is greater than the sum of the squares on the two other sides, say with sides of 2, 3 and 4 inches. What relation does the angle opposite the greatest side bear to a right angle? Measure it with protractor
13. Construct a triangle with sides of 30, 40 and 55 millimetres (552 >302 +40). What sort of angle is that opposite the greatest side?
14. Describe a triangle such that the square on one side is less than the sum of the squares on the two other sides, say with sides of