大三程度常識題舉隅

甲部 自由回應題

問題：

每個女性向文學作品的角色們都有唔少性格密碼讓心水清的讀者自行發掘。以下是其中一個：

男人似自己外公，女人似自己祖母

試指出上述性格密碼所蘊含的科學原理。

Section 3: Problem Solving

Given AC=7, AB=24, BC=25,

and angle ADC = 45 degree,

compute the area of triangle ADC.

Solution:

By Pythagoras theorem,

BAC is a right-angle triangle.

Denote angle BAD= x,

angle ABD = 45 degree - x,

angle ACD = 45 degree + x,

angle CAD = 90 degree - x

Now

sin (angle CAD) = cos(x)

sin (angle ACD) = 24/25

= cos(90 degree - 45 degree - x)

= cos (45 degree - x)

sin (angle ABD) = 7/25

= cos(90 degree - 45 degree + x)

= cos (45 degree + x)

cos (angle CAD) = sin(x)

cos (angle ACD) = 7/25

= sin (45 degree - x)

cos (angle ABD) = 24/25

= sin (45 degree + x)

Now, use a calculator to get angle ABD:

Then draw a line from a point E on line AC such that AB//DE. Now angle ADE = x, and angle CDE = 45 degree - x

AE = DE tan(x),

CE = DE tan(90 degree - x),

AE + CE = 7 = AC

then 7 = DE [tan(90 degree - x) + tan(x)]

Use a calculator to conpute DE then.

Area of the required triangle

= AC • DE / 2

Now you see that within junior high math curriculum alone, we can still derive the answer easily. For DSE level, simply use sine law when x is obtained.

Erratum:

AE = DE tan(x),

CE = DE tan(45 degree - x),

AE + CE = 7 = AC

then 7 = DE [tan(45 degree - x) + tan(x)]

Section C: Problem Solving