中三升中四暑期功課數學題

1. In the figure, the angles of depression of points A and B from the top of a building CD are 30 and 48 respectively. If A and B lie on the same plane and are 150 m apart, find the height of the building correct to 4 significant figures.
Let CD be h m and BD be x m.
∠CAD=30 degree and ∠CBD=48 degree
In rt∠△CDB, tan∠CBD=h/x
tan 48 degree=h/x
x=h/tan48 degree . . . . . . . . . (1)
In rt∠△CDA, tan∠CAD=h/(150+x)
tan 30 degree=h/(150+x)
150+x=h/tan 30 degree
x=h/tan 30 degree-150 . . . . . (2)
Compare (1) and (2) :
h/tan48 degree=h/tan 30 degree-150
h/tan 30 degree-h/tan48 degree=150
h(1/tan 30 degree-1/tan48 degree)=150
h=150/(1/tan 30 degree-1/tan48 degree)
=150/(1.73205-0.90040)
=150/0.83165
=180.4
The height of the building correct to 4 significant figures is 180.4 m.

2. In the figure, P(2, 5), Q(5, 1), R(1, -2) and S are the vertices of a square. Find the coordinates of S.
Slope of PQ : (5–1)/(2–5)=–4/3
Slope of QR : [1–(–2)]/(5–1)=3/4
PS//QR, equation of PS : y–5=(3/4)(x–2)
4y–20=3x–6
3x–4y+14=0 . . . . . (1)
SR//PQ, equation of SR : y–(–2)=(–4/3)(x–1)
3y+6=–4x+4
4x+3y+2=0 . . . . . (2)
(1)x3+(2)x4 : 25x+50=0
x=–2
Put x=–2 into (1) : 3(–2)–4y+14=0
y=2
The coordinates of S are (–2,2).