Heights and Distances - Worksheet 1
1.
In the figure O is the centre of the circle.
AO = OC = r and AB = CB = d.
∠ABC = 1800 – x and ∠AOC = 2 x.
O
r 2x
r Prove that: 𝑑 = 𝑟√2(1 − 𝑐𝑜𝑠𝑥)
A C
d 180 - x
d
B
2.
In ΔABC, AM is a median. (ie BM = MC).
A
∠BAM = x, ∠B = y and ∠C = 300.
x
(a) Determine an expression for the
length of AM in terms of x, y and d.
(b) Hence show that:
y 30 𝑑 2𝑠𝑖𝑛2 𝑦
𝐴𝐶 = 2 (𝑠𝑖𝑛2𝑦 + )
B M C 𝑡𝑎𝑛𝑥
d
3.
AC represents a vertical tower.
A
ΔBCD lies in the horizontal plane.
∠BCD = 900 – x, ∠BDC = 2 x and ∠BAC = x
x AB has a length of 2 units.
(a) Show that BD = 1 unit
3
2 (b) If AD = √3, calculate the size of
∠ABD without using a calculator.
C
90 - x
2x
B D
1
1.
In the figure O is the centre of the circle.
AO = OC = r and AB = CB = d.
∠ABC = 1800 – x and ∠AOC = 2 x.
O
r 2x
r Prove that: 𝑑 = 𝑟√2(1 − 𝑐𝑜𝑠𝑥)
A C
d 180 - x
d
B
2.
In ΔABC, AM is a median. (ie BM = MC).
A
∠BAM = x, ∠B = y and ∠C = 300.
x
(a) Determine an expression for the
length of AM in terms of x, y and d.
(b) Hence show that:
y 30 𝑑 2𝑠𝑖𝑛2 𝑦
𝐴𝐶 = 2 (𝑠𝑖𝑛2𝑦 + )
B M C 𝑡𝑎𝑛𝑥
d
3.
AC represents a vertical tower.
A
ΔBCD lies in the horizontal plane.
∠BCD = 900 – x, ∠BDC = 2 x and ∠BAC = x
x AB has a length of 2 units.
(a) Show that BD = 1 unit
3
2 (b) If AD = √3, calculate the size of
∠ABD without using a calculator.
C
90 - x
2x
B D
1
