Mathematics (O-Level)

Mathematics (O-Level)

Question:In the following fig. XY is parallel to BC. Prove that the triangles ABC and AXY are similar, and hence find two expressions equal to `(XY)/(BC)`. 

figure Line 
AnswerSince XY is parallel to BC,
              `hat(X)=hat(B)` (corresponding).
Similarly  `hat(Y)=hat(C)`.
The triangle `{("AXY"),("ABC"):}` are equiangular and therefore similar.

`:. (XY)/(BC)=(AX)/(AB)=(AY)/(AC)`

(Notice that any ratio of lengths on AB is equal to the corresponding ratio of lengths on AC. For example, `(AX)/(XB)=(AY)/(YC)`. The ratio of the transversals, XY and BC is, however, equal only to the ratio of a side of the small triangle AXY to the corresponding side of the large triangle ABC.)

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Question:The sides of the triangle ABC are 3 cm, 3 cm and 5 cm. The sides of the triangle XYZ are 9 cm, 12 cm and 15 cm. If the area of the  triangle ABC is 6 `cm^2` , find the area of the triangle XYZ is 54 `cm^2` . 
AnswerEach side of the triangle XYZ is three times the corresponding side of the triangle ABC. The area of XYZ is therefore nine times as great as the area of ABC. Therefore the area of the triangle XYZ is 54 `cm^2` .

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Question:A bottle holds 1 litre. How much would a similar bottle, in which each length is doubled, hold? 
AnswerThe ratio of the linear lengths is 2:1.
   The ratio of the volumes is `2^3:1^3` or 8:1.
The second bottle therefore holds 8 litres.

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Question:In the triangle ABC, the angle B=`90^0`, the angle A=`42^0` and AC=10 cm. Find the length of BC. 
AnswerConsidering the angle A, the side BC is the opposite side and AC is the hypotenuse. Therefore sin `42^0`=`BC/10`
`:.` CB=10 sin `42^0` cm = `10 xx 0.6691 cm = 6.691 cm`
           = 6.69 cm (correct to 3 sig. fig.).

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Question:In the triangle ABC, the angle B=`90^@` and the angle A=`38^@`, Given that BC=4 cm, calculate the Length of AB. 
AnswerThe tangent of the angle A=`(BC)/(AB)` . The unknown side is the denominator of the fraction. Therefore consider the angle C.
The angle C = `90^@` - `38^@` = `52^@`
                                 `:. tan 52^@ = (AB)/4`
`:.` 4 tan `52^@` cm = `4 xx 1.2799 cm`
                                    = 5.1196 cm
                                    = 5.12 cm (correct to 3 sig. fig.).

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Question:In the following fig. XY is parallel to BC. Prove that the triangles ABC and AXY are similar, and hence find two expressions equal to `(XY)/(BC)`. figure Line

Question:The sides of the triangle ABC are 3 cm, 3 cm and 5 cm. The sides of the triangle XYZ are 9 cm, 12 cm and 15 cm. If the area of the triangle ABC is 6 `cm^2` , find the area of the triangle XYZ is 54 `cm^2` .

Question:A bottle holds 1 litre. How much would a similar bottle, in which each length is doubled, hold?

Question:In the triangle ABC, the angle B=`90^0`, the angle A=`42^0` and AC=10 cm. Find the length of BC.

Question:In the triangle ABC, the angle B=`90^@` and the angle A=`38^@`, Given that BC=4 cm, calculate the Length of AB.