By : Darto, SMP N 4 Pakem Media and Video Problem -2 Problem -1.
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Transcript of By : Darto, SMP N 4 Pakem Media and Video Problem -2 Problem -1.
By : Darto, SMP N 4 Pakem
Media and Video
Problem -2
Problem -1
Angl
es in
a T
riang
les
PROBLEM -11.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)
Cek, please
Bisa Tidak
4, 7, 8
5, 6, 10
7, 15, 6
8, 10, 19
14
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
15
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
16
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>155+15 >12 or 20>1212+15 >5 or 27>5
17
The measures of two sides of a triangle are 15 and 8. Between what two numbers is the third side.
X
15+8 > X
15+X > 8
8+X > 15
STANDARD 6
23 > XX < 23
15+X > 8-15 -15 X > -7
8+X > 15-8 -8
X > 7
0 5 10 15 20 25x
-5-10
-15
-20
x
x
x
-7
7
23
X | 7<X<23
815
The third side will be any value between 7 and 23.
7 23
18
If a triangle has sides of measure x, x+4, 3x-5, find all possible values of x
(X+4)+(3X-5) > X
(X+4 )+X > (3X-5)
X
X+4
3X-5
4X -1 >X-4X -
4X -1 >-3X-3 -3.3 <X
X>.3
2X +4 > 3X-5-2X -
2X 4 > X-5+5 +59 >
XX < 9
Sign (>) changes when dividing by (-3)
0 5 10 15 20 25x
-5-10
-15
-20
x
x
x
9
3
.3
(3X-5) +X > (X+4 )4X – 5 > X
+4 -X -X3X – 5 > 4 +5 +5
3X > 93 3X > 3
3 9X | 3<X< 9
19
If one side of a triangle is the longest then
A
B
C
20
If one side of a triangle is the longest then
The opposite angle to this side is the largest
A
B
C
21
And the angle opposite to the shortest side
A
B
C
22
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
1. Do you remember about acute angle2. Observe the size all of angle in the triangle
bellow
1. Do you remember about obtuse angle
2. Observe the size all of angle in the triangle bellow
THE KINDS OF TRIANGLEBASE ON THE SIZE ANGLEIII. 1. Do you remember about right angle
2. Observe the size all of angle in the triangle bellow
Problem -3Determine the kind of triangle bellow if
1. The angle are : 65, 75, 802. 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 TRIANGLEBASE ON THE LENGTH SIDEI. 1. Observe the length of all side in the triangle
bellow
THE KINDS OF TRIANGLEBASE ON THE LENGTH SIDEII. 1. Observe the length of all side in the triangle
bellow
THE KINDS OF TRIANGLEBASE ON THE LENGTH SIDEIII. 1. Observe the length of all side in the triangle
bellow
RIGHT TRIANGLES
1. Recall Pythagorean theorem2. Indentify The kinds of Triangle by using
Pythagorean theorem3. The kinds of Triple Pythagorean number and
its expectation 4. The specific side proportion of right triangle
30°-60°-90° TRIANGLE
45°-45°-90° TRIANGLE
PROBLEM 1
PROBLEM 2
PROBLEM 4
PROBLEM 5
PROBLEM 6
PROBLEM 3
2
2
2
60°
60°
60°60°
30°
2
2
1
1
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°
30°
1
2
2
1
60°
30°
12
z
2 = z + 12 22
4 = z + 12
-1 -13 = z2
3 = z 2
z = 3
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.
Can we generalize this result for all 30°-60°-90° right triangles? Click to find out…
60°
30°
12
3
(.5)(.5)
(.5)
(2)
(2)
(2)
(s)(s)
(s)
12
3
60°
30°
1
2
3
60°
30°
12
3
60°
30°
7. Is this true for a triangle that is twice as big?
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? Click to find out…
2
3s
s
s
60°
30°
3
In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.
THEOREM 8-7
Find the values of the variables. Round your answers to the nearest hundredth.
x
y
30°
50
90°-30°=60°
60°
2
3s
s
s
60°
30°
2x = 50
2x = 50
2 2
x =25
y = x 3
y = 25 3
OR
Is this 30°-60°-90°?
Then we know that:
y = 43.30
60°
30°
90°-60°=30°
x
y 90
90 = x 3
90 = x 3
3 3
90
3= x
3
3 .
= x390
3( ) 2
390
3= x
330 = x
OR
2x = y
330( ) = y2
= y360
y= 360
OR
330 x=
Find the values of the variables. Round your answers to the nearest unit.
Is this a 30°-60°-90°? y =104
x = 52
2
3s
s
s
60°
30°
60°
30°
90°-60°=30°
x
y
30
30 = x 3
30 = x 3
3 3
30
3= x
3
3 .
= x330
3( ) 2
330
3= x
310 = x
2x = y
310( ) = y2
= y320
y= 320
310 x=
Find the values of the variables. Find the exact answer.
Is this a 30°-60°-90°?
2
3s
s
s
60°
30°
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
1
1
1
45°
45°
45°
45°
45°
45°
1
1y
y = 1 + 12 22
y = 1 + 12
y = 22
y = 22
y = 2
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…
45°
45°
1
12
(.5)
(.5)
(.5)
(1.5)
(1.5)
(1.5)
s
ss
1
1
2
45°
45°
1
1
2
45°
45°
5. Let’s draw a triangle half the size of the original.
6. Let’s draw a triangle one and a half the size of the original.
7. Let’s draw a triangle S times the size of the original.
Click to see our findings…
s
s
s 2
In a 45°-45°-90° triangle, the hypotenuse is times as long as a leg.
THEOREM 8-6
2
45°
45°
45°
90°-45°=45°
45°
x
y
36
If y = x
36 = x 2
36 = x 2
2 2
36
2= x
2
2 .
= x236
2( ) 2
236
2= x
218 = x
218x =OR
218y =
OR
then
Find the values of the variables. Round your answers to the nearest tenth.
s
s s 2
45°
45°
Is this a 45°-45°-90°?
x = 25.5
y = 25.5
45°
90°-45°=45°
45°
x
y
42
If y = x
42 = x 2
42 = x 2
2 2
42
2= x
2
2 .
= x242
2( ) 2
242
2= x
221 = x
221x =
221y =then
Find the values of the variables. Give an exact answer.
s
s s 2
45°
45°
Is this a 45°-45°-90°?
45°
90°-45°=45°
x
21
y
45°
x = 21
2 xy=
221y =
Find the values of the variables. Give the exact answer.
Is this a 45°-45°-90°?
s
s s 2
45°
45°
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