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Chapter 5 Introduction to Trigonometry: 5.4 Modelling with
Similar Triangles
Humour Break
5.4 Modelling with Similar Triangles
Goals for Today:• Apply what we have learned about similar
triangles to some word problems
5.4 Modelling with Similar Triangles• Ex. 1 – Rachel is standing beside a lighthouse
on a sunny day. Rachel is 1.6m tall and the sun casts a shadow 4.8m long (on the ground). At the same time, the sun casts a shadow 75m long (on the ground). How tall is the lighthouse?
5.4 Modelling with Similar Triangles
5.4 Modelling with Similar Triangles
5.4 Modelling with Similar Triangles
• AB is ll (parallel) to DE (same sun’s rays)• AC is on the same line as DF (both flat on ground)
• ∠BAC = EDF (same )∠ ∠• ∠ACB = DFE (both 90°)∠• ∠ABC = DEF (180° ∆ rule)∠• Therefore, ∆ABC ~ ∆DEF
5.4 Modelling with Similar Triangles
mx
x
x
xx
DF
AC
DE
AB
258.4
120
8.4
8.4
1208.4
)6.1)(75(8.4
6.1
75
8.4
• Plug in the known values
• Cross-multiply
• Simplify & solve for x
Since ∆ABC ~ ∆DEF
5.4 Modelling with Similar Triangles• Ex. 2 – A 3.6 m ladder is leaning against a wall
with its base 2 m from the wall. A shorter 2.4 m ladder is placed against the wall parallel to the longer ladder.
• (a) How far will the smaller ladder reach up the wall?
• (b) How far will the smaller ladder’s base be from the wall?
5.4 Modelling with Similar Triangles
5.4 Modelling with Similar Triangles
5.4 Modelling with Similar Triangles
• AB is ll (parallel) to DE (ladder at same )∠• ∠ABC = DEC (F pattern)∠• ∠BAC = EDC (F pattern)∠• ∠ACB = DCE (common 90°)∠• Therefore, ∆ABC ~ ∆DEC
5.4 Modelling with Similar Triangles• For (a), we first need to find out how far the
larger ladder reaches up the wall using the pythagorean theorem…
5.4 Modelling with Similar Triangles• c² = a² + b² (Pytharorean theorem)• DE² = DC² + EC² (Substitute triangle sides)• 3.6² = DC² + 2² (Use algebra to solve)• 12.96 = DC² + 4• 12.96 – 4 = DC²• DC²=8.96 (Take √ of both sides)• √DC²= √8.96• DC≈3 m
5.4 Modelling with Similar Triangles
5.4 Modelling with Similar Triangles
mAC
AC
AC
AC
ACDC
AC
DE
AB
26.3
2.7
6.3
6.3
2.76.3
)3)(4.2(6.336.3
4.2
• Plug in the known values
• Cross-multiply
• Simplify & solve for AC
Since ∆ABC ~ ∆DEC (a) to find AC (how far smaller ladder reaches up the wall)…
5.4 Modelling with Similar Triangles• How far will the smaller ladder be from the
base of the wall?• Here, we have EC, so we can set-up a ratio
using our similar triangle relationship
5.4 Modelling with Similar Triangles
mBC
BC
BC
BC
BCEC
BC
DE
AB
3
11
6.3
8.4
6.3
6.3
8.46.3
)2)(4.2(6.326.3
4.2
• Plug in the known values
• Cross-multiply
• Simplify & solve for BC
Since ∆ABC ~ ∆DEC to find BC (how far smaller ladder base will be from the wall)…
Homework
• P.474, #1-4 & 6-13
