Exploring
Right
Triangles
Dora leaves for a long hike.
She walks 6 miles north.
Then, she hikes 4 miles west.
She then turns and goes 2 miles due south.
How far is she from her place of origin?
We can solve this the old school way…
using the Distance Formula.
She starts at the origin and stops at the point (-4, 4).
 x2  x1    y2  y1 
2
 4  0    4  0 
2
 4    4 
2
2
2
2
16  16  32  4 2
Or we can try some fancy stuff…
Insert a
horizontal
line to
create a
right
triangle.
How about the Pythagorean Theorem?
4
2
4
a b  c
2
2
2
4 4  x
2
2
16  16  x
2
6
32  x
2
4
16 2  x
4 2x
2
2
2
Not fancy enough?
4
2
2
4
6
4
Did you
notice
that both
legs of the
right
triangle
measure
4?
4
That
means
2
2
45·45·90
right
4
6
triangle!!!
4
45°
4
2
x, x, x 2
4 , 4 , ...
2 4, 4, 4 2
4
6
4
Or we can try some SUPER, SUPER fancy stuff…
TRIG!!!
sin4 5  =
4
4
x
OR
cos4 5  =
4
Solving for x,
4
x
4
4
x=
OR x =
, x  5 .6 5 6 9
sin4 5
cos4 5
WAIT A MINUTE…
That’s not the same answer…
x  5 .6 5 6 9 ?
4
And using a calculator...
4 2 ?
4  2  5 .6 5 6 9
4
To recap and review…
x  4 2  5.65697
4
4
How do I know which to use?…
Distance
Formula
Pythagorean
Theorem
When you know the
coordinates of the
endpoints of the
segment.
When you have the
lengths of two sides.
2
𝑎2 + 𝑏 2 = 𝑐 2
𝑥2 − 𝑥1
+ 𝑦2 − 𝑦1
2
Special Right
Triangles
When it fits one of
the two patterns:
45*45*90
or
30*60*90.
𝑥, 𝑥, 𝑥 2
OR
x, x 3, 2 x
TRIG
Ratios
When you know the
angle measure and
at least one other
side measure.
SOHCAHTOA