Media and Video
By : Darto, SMP N 4 Pakem
Problem -1
Problem -2
Angles in a Triangles
PROBLEM -1
1.Determine the angle in the triangle isn’t known yet if
given two angle :
a. 23, 67, …
b. 37, 84, …
2. Determine the value of x
if the angle in the triangle are
a. 4x, 5x+6, 9x -6
b. 2x, 3x, 5x
EXERCISES-1
EXERCISES-2
Conclusion
Pasangan seg.
Garis(cm)
4, 7, 8
5, 6, 10
7, 15, 6
8, 10, 19
Cek, please
Bisa Tidak
The sum of the lengths of any two sides of a triangle is
greater than the third side.
5
12
15
5+15 >12 or 20>12
14
The sum of the lengths of any two sides of a triangle is
greater than the third side.
5
12
15
12+15 >5 or 27>5
15
The sum of the lengths of any two sides of a triangle is
greater than the third side.
5
12
15
5+12 >15 or 17>15
5+15 >12 or 20>12
12+15 >5 or 27>5
16
STANDARD 6
The measures of two sides of a triangle are 15 and 8. Between
what two numbers is the third side.
15
x
7
8
x
-7
23
- -15 -10
20
X
15+8 > X
23 > X
X < 23
0
5
10 15
7
20 25
23
X | 7<X<23
15+X > 8
8+X > 15
15+X > 8
-15
-15
8+X > 15
-8
-8
X > -7
-5
X>7
The third side
will be any
value between 7
and 23.
17
x
x
If a triangle has sides of measure x, x+4, 3x-5, find all possible
values of x
(3X-5) +X > (X+4 )
4X – 5 > X +4
X+4
3X-5
-X
-X
3X – 5 > 4
+5 +5
X
3X > 9
(X+4)+(3X-5) > X (X+4 )+X > (3X-5)
4X -1 >X
2X +4 > 3X-5
-2X
-2X
-4X
-4X
4 > X-5
-1 >-3X
-3 -3
+5
+5
Sign (>) changes
when dividing by
9>X
.3 <X
(-3)
X<9
X>.3
X | 3<X< 9
3
3
X>3
x
3
x
9
.3
- -15 -10
20
-5
5
0
3
10 15
20 25
9
18
x
x
If one side of a triangle is the longest then
A
B
C
19
If one side of a triangle is the longest then
A
B
C
The opposite angle to this side is the
largest
20
B
A
C
And the angle opposite to the shortest side
21
The sum of the lengths of any two sides of a triangle is
greater than the third side.
5
12
15
5+12 >15 or 17>15
22
Do you remember about acute angle
2. Observe the size all of angle in the triangle bellow
1.
1. Do you remember about obtuse angle
2. Observe the size all of angle in the
triangle bellow
THE KINDS OF TRIANGLE
BASE ON THE SIZE ANGLE
III. 1. Do you remember about right angle
2. Observe the size all of angle in the triangle bellow
Problem -3
 Determine the kind of triangle bellow if
1. The angle are : 65, 75, 80
2. The angle are : 25, 60, 95
3. The angle are : 54, 56, 70
4. Two angle are : 73, 34,
5. The proportion of angle is 3 : 4 : 5
6. The proportion of angle is 2 : 3 : 4
7. The angle is 6x, 2x + 3, 4x +9
PROBLEM-3
THE KINDS OF TRIANGLE
BASE ON THE LENGTH SIDE
I.
1. Observe the length of all side in the triangle bellow
THE KINDS OF TRIANGLE
BASE ON THE LENGTH SIDE
II.
1. Observe the length of all side in the triangle bellow
THE KINDS OF TRIANGLE
BASE ON THE LENGTH SIDE
III. 1. Observe the length of all side in the triangle bellow
RIGHT TRIANGLES
1. Recall Pythagorean theorem
2. Indentify The kinds of Triangle by using Pythagorean
theorem
3. The kinds of Triple Pythagorean number and its
expectation
4. The specific side proportion of right triangle
30°-60°-90° TRIANGLE
PROBLEM 1
PROBLEM 2
PROBLEM 3
45°-45°-90° TRIANGLE
PROBLEM 4
PROBLEM 5
PROBLEM 6
60°
60°
2
2
1
30°
60°
2
1
60°
2
2
1. An equilateral triangle is also equiangular, all angles are the same.
2. Let’s draw an Altitude from one of the vertices. Which is also a Median and Angle
bisector.
3. The bisected side is divided into two equal segments and the bisected angle has
now two 30° equal angles.
How is the right angle that was formed? Click to find out
60°
2
60°
1
2
1
30°
30°
z
1
2
2
4. The triangle is divided into 2 right angles with
acute angles of 30° and 60°
5. Let’s draw the top triangle and label the
unknown side as z.
6. Let’s apply the Pythagorean Theorem to find the
unknown side.
2
2 = z +1
4 = z 2+ 1
-1
-1
2
3=z
3= z
2
z= 3
Can we generalize this result for all 30°-60°-90° right triangles? Click to find out…
2
60°
60°
2
(2) 2
1
(2) 1
30°
3
30°
(2)
(.5) 1
60°
(.5) 2
3
30°
(.5)
60°
3
7. Is this true for a triangle that is twice as big?
2 (s)
(s) 1
8. Is this true for a triangle that is half the
original size?
9. What about a triangle that is “s” times bigger or
Smaller?
30°
(s)
3
Click to find out…
THEOREM 8-7
60°
2s
s
30°
s
3
In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter
leg, and the longer leg is 3 times as long as the shorter leg.
Find the values of the variables. Round your answers to the nearest hundredth.
y
2x = 50
30°
2x = 50
2
2
x
50
60°
x =25
Is this 30°-60°-90°?
90°-30°=60°
y =x 3
Then we know that:
60°
y = 25 3
2s
OR
s
y = 43.30
30°
s
3
Find the values of the variables. Round your answers to the nearest unit.
90 = x
3
90 = x
3
30°
y
3
2 ( 30 3
.3
90
3
60°
90 3
x
(
3 )
Is this a 30°-60°-90°?
90°-60°=30°
90 3
3
60°
2s
30 3
s
30°
3
) =y
3
90
3
s
2x = y
60 3 = y
=x
y= 60 3
OR
=x
2
y =104
=x
=x
x= 30 3
OR
x = 52
Find the values of the variables. Find the exact answer.
x
30 = x
3
30 = x
3
2x = y
60°
3
y
30
30°
.3
3
30 3
(
3 )
Is this a 30°-60°-90°?
90°-60°=30°
30 3
3
60°
2s
10 3
s
30°
s
3
) =y
3
30
3
2 ( 10 3
20 3 = y
=x
=x
2
=x
=x
x= 10 3
y= 20 3
1
1
1
1
1. Let’s draw a diagonal for the square above. The diagonal bisects the right angles of the
square.
What kind of right triangles are form? Click to find out…
1
45°
45°
45°
1
y
1
1
45°
45°
45°
1
1
2. The triangles are 45°-45°-90°
3. Let’s draw the bottom triangle and label the
hypotenuse as y
4. Let’s apply the Pythagorean Theorem to find
the hypotenuse.
Can we generalize our findings? Click to find out…
2
2
y = 1 +1
2
y =1+1
y2 = 2
y2 = 2
y= 2
2
(.5) 2
45°
s 2
1s
45° (.5) 1
45°
(.5) 1
5. Let’s draw a triangle
half the size of the
original.
45°
s1
45°
(1.5)
2
6. Let’s draw a triangle
one and a half the size of
the original.
(1.5) 1
7. Let’s draw a triangle
S times the size of the
original.
45°
(1.5) 1
Click to see our findings…
THEOREM 8-6
45°
s 2
s
45°
s
In a 45°-45°-90° triangle, the hypotenuse is 2 times as long as a leg.
Find the values of the variables. Round your answers to the nearest tenth.
36 = x 2
36 = x 2
45°
36
x
2
2
2
.2
If y = x
36
2
=x
45°
36 2
y
(
Is this a 45°-45°-90°?
90°-45°=45°
36 2
45°
s 2
2 )
2
18 2
s
=x
2
=x
=x
OR
s
y = 18 2
OR
y = 25.5
x = 18 2
45°
then
x = 25.5
Find the values of the variables. Give an exact answer.
42 = x 2
45°
42 = x 2
42
2
2
2
.2
x
If y = x
42
2
=x
45°
42 2
y
(
Is this a 45°-45°-90°?
90°-45°=45°
42 2
45°
s 2
2 )
2
s
21 2
=x
2
=x
=x
x = 21 2
45°
s
then
y = 21 2
Find the values of the variables. Give the exact answer.
x = 21
45°
y= x
y
x
2
y = 21 2
45°
glas-Opera. mpg
21
Is this a 45°-45°-90°?
90°-45°=45°
45°
s 2
45°
s
s