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Head First 2D Geometry by Dawn Griffiths, Stray

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mind the gap
What’s the longest the rope can be without going below the
safety line?
4m + 2m = 6m
Safety line
The floor
4m
2m
2m
T
his 2m is below the safety line, so
not included
Onl
y add up the
heights ABOVE the
safety line.
So, how far can you swing on a six-meter rope?
The gap between the platforms is the base of a triangle, with the rope
making up two of the sides joined at the point where the rope is fixed at the
top. So the distance of the gap is the same as the length of the base of the
triangle.
W
e need t
o find the
dis
tance of this gap.
T
his angle isn’t
a r
ight angle.
Safety line
The floor
4m
2m
d
142 Chapter 3
the pythagorean theorem
Well, the rope doesn’t change
length, does it? So I guess that gives
us two sides of the triangle, but we
can’t use the Pythagorean Theorem
to find the missing side without a
right angle, can we?
How can you make the swing problem into a
right triangle problem so that you can use the
Pythagorean Theorem to find length d?
Safety line
The floor
4m
2m
d
6m
That’s right—the Pythagorean Theorem
only finds missing sides for right
triangles.
So, when you’re faced with a triangle without a right
angle you’ve got two options: find something else in
your Geometry Toolbox to solve the problem, or see if
you can somehow turn your non-right triangle into a
right triangle (or triangles!)
you are here 4 143
BISECTS this line.
Bisec
t
create your own right triangles
How can you make the swing problem into a right triangle problem
so that you can use the Pythagorean Theorem to find d?
Solution
The base of the triangle is horizonal, so
a vertical line down from the top of the
triangle splits it into two equal right
triangles.
Safety line
The floor
4m
2m
d
6m
You can split an isoceles triangle into two
congruent right triangles
You can split a triangle into two right triangles by drawing an
altitude—a line which joins the top of the triangle to the base,
perpendicular to the base. An isoceles triangle has two sides and two
angles the same, so the two triangles created are congruent.
6m 6m6m 6m
W
Two sides the same
Two angles the same
T
his line is called
an a
lti
tude.
T
he alti
tude
BISECTS
this angle.
T
he alti
tude
Chop int
o t
wo
equal parts.
W
144 Chapter 3
the pythagorean theorem
BE the Rope Swing
Your job is to play like you’re the 6m
rope. Use the Pythagorean Theorem to
work out how far across a gap someone
could swing on you.
Q:
I get that the rope length doesn’t
change, so the sides are equal, but how
did we know that the bottom two angles
are equal?
A:An isoceles triangle has two sides
equal but also two angles equal—always.
So—if you see that the sides are the same
you know the angles are the same, and the
other way around.
Q:
How did we know that the altitude
Q:
So, which triangle does the
would be vertical? altitude belong to?
A:The altitude is always perpendicular A:Both! The altitude is the shared side
to the base that it’s drawn on, so if that of the two identical (congruent) triangles it
base is horizontal (as it is in this case), creates in this case. So, it belongs to both
then the altitude must be vertical. of them.
you are here 4 145

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