Section 9 - The Square and the Equilateral Triangle - Willets Geometry

Section 9 - The Square and the Equilateral Triangle:
1. The diagonal of a square separates the square into two 45°-45°-90° Triangles.
The diagonal of a square is the same as
the hypotenuse of a right triangle
and
a side of the square is the same as
a leg of a right triangle.
1
Side of Square Leg of DCB hypotenuse
2
Diagonal of Square Hypotenuse of DBC leg
2. The altitude of an equilateral triangle separates
the triangle into two 30°-60°-90° triangles. Since
each angle of an equilateral triangle contains 60°,
the altitude of the triangle is opposite a 60° angle
and the side of the equilateral triangle is the
hypotenuse of the right triangle.
Side AC of Equilateral ABC Hypotenuse of 30 60 90
ADC
1
Altitude CD Leg opposite 60 hypotenuse 3
2
1. Find the diagonal of a square whose side is 8.
A
D
x 45
45
B
8
C
A B
D
45°
Examples:
side
A
45°
60
Since the diagonal of the square is the same
as the hypotenuse of the right triangle, we use the
formula: leg 2
C
So, x = 8 2
C
D
altitude
B
2
2

2. Find the side of a square whose diagonal is 12.
3. Find the altitude of an equilateral triangle whose side is 12.
A
12
60
A
D
C
30
12
45
x
45
D B
B
C
x
Since the side of the square is the same as the leg of the
right triangle, we use the formula:
1
2 hypotenuse
So, x = 6 2
Since ABC is an equilateral triangle, mA60 .
AC , the left side of the equilateral triangle
is also the hypotenuse of ADC .
In this triangle, “x” is the leg opposite the 60 angle.
We use the formula:
So, x = 6 3
2
1
2 hypotenuse
3

Assignment: Section 9
1. Find the diagonal of a square whose side is 10.
2. Find the side of a square whose diagonal is 16.
3. Find the altitude of an equilateral triangle whose side is 6
4. Find the side of a square whose diagonal is 12.
5. Find the diagonal of a square whose side is 7.
6. Find the altitude of an equilateral triangle whose side is 18.
7. Find the diagonal of a square whose side is 14.
8. Find the altitude of an equilateral triangle whose side is 16.
9. Find the side of a square whose diagonal is 8.
10. Find the altitude of an equilateral triangle whose side is 4.
11. Find the diagonal of a square whose side is 3.
12. Find the side of a square whose diagonal is 14.