Distance Survey
By Leighton McIntyre
Goal: to calculate distance given obstructions in path
Problem
In conducting a land survey, the following problem
arose. There were two points A and B along a road and
points C and D off the road along the respective
perpendiculars to the road at A and B. There were
buildings on the property that prevented direct
measurement of the distances along BD and AC.
Measurements, however, could be made for AD, BC,
and AB as follows:
AB = 240 yards
AD = 260 yards
BC = 300 yards
A light pole is to be installed at point E, the
intersection of CB and AD. How far will the pole be
from the road? That is, what is the distance EF?
Again, an existing building blocks direct
measurement.
Solution 1
AC ⊥ AB , BD ⊥ AB, ⇒ AC //BD
then BC and AC are both transversals to AC and BD.
The following angle congruencies hold:
<ACE ≈ <BED , <ACB ≈ <CBD, <CEA ≈ <DEB
∆ACE ~ ∆ BED
AC2 = CB2 - AB2 = 3002 - 2402 = 32400
hence AC = 180
BD2 = AD2 - AB2 = 2602 - 2402 = 10000
hence BD = 100
Ratio of sides of ∆ ACE to ∆ BED 180:100 or 9:5
CE: BE = 9:5
CE = 9 BE and CE +BE = BC
5
€
Hence BE =
5
BC
14
=
5
*300
14
=
750
7
AE = 9 DE and AE + DE = AD
5
€
€
9
AD
14
Hence AE =
€
=
€
9
*260
14
=
1170
7
Now using Herons Formula we can calculate the area
of triangle
ABE€ because€we have the three sides and
€
then find height EF
Area ΔABE =
s(s − a)(s − b)(s − c)
Where a = 240, b =
€
Area ΔABE =
1170
,
7
c=
750
7
,s=
a+b+c
= 1800
2
7
378000
1800 1800
1800 1170 1800 750
(
− 240)(
−
)(
−
)=
7
7
7
7
7
7
49
€
€
€
€
€
€
Now we know using the base of the triangle as 240
and EF as height in formula
Area ΔABE = 1 base*height
2
€
378000 1
= (240) EF
49
2
€
EF = 378000 = 3150 =
49 *120
49
€
€
€
Solution 2
64.29
AC ⊥ AB , BD ⊥ AB, EF ⊥ AB ⇒ AC //BD// EF
From solution 1, we calculated that AC = 180 and BD
= 100
Denote by x the distance EF. Denote by y the distance
FB
Consider ∆ ABC and ∆ FBE are similar by AA
similarity.
That is <CAB ≈ <EFB, by right angles; <FBE ≈ <FBE,
by reflexive property.
Consider ∆ ABD and ∆ AFE are similar by AA
similarity.
That is <ABD ≈ <AFE, by right angles; <FAE ≈ <FAE,
by reflexive property.
We have the following proportions
€
CA
AB
=
EF
FB
⇒
€ DA
=
€
EF
FA
⇒
AB
x))
€⇒
€
180
240
=
x
y
⇒3 =
4
€100 €
€
= x
240 − y
240
⇒ 12x + ( 20 x))= 1200
3
⇒
56
x
3
€
€
= 1200
⇒ x = 64.29
€
⇒ y = 4x
3
€
⇒5 =
€
12x€ = 1200
– ( 20€x)) €
3
x
y
12
x
⇒
240 − y
12x = 5(240 – ( 4
3
€