NAME ______________________________________________ DATE

____________ PERIOD _____

7-1

Practice

(Average)

Geometric Mean
Find the geometric mean between each pair of numbers to the nearest tenth. 1. 8 and 12 2. 3 7 and 6 7 3. 4 and 2 5

96

9.8

126

11.2

8 5

1.3

Find the measure of each altitude. State exact answers and answers to the nearest tenth. 4. T U

5. J
V

6

M
17

5 A

12

L

K

Answers

60

7.7

102

10.1

Find x, y, and z. 6.
y
8

7.
23

25

6

x z

z

x

(Lesson 7-1)

y

184 713
8.
y
2 3

13.6; 26.7

248

15.7;

114 475
9.
z

10.7; 21.8

150

12.2;

10

z

x

x

20

y

4.5;

13

3.6; 6.5

15; 5;

300

17.3

10. CIVIL ENGINEERING An airport, a factory, and a shopping center are at the vertices of a right triangle formed by three highways. The airport and factory are 6.0 miles apart. Their distances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service road will be constructed from the shopping center to the highway that connects the airport and factory. What is the shortest possible length for the service road? Round to the nearest hundredth. 2.88 mi

NAME ______________________________________________ DATE

____________ PERIOD _____

7-2
Find x. 1.
x

Practice

(Average)

The Pythagorean Theorem and Its Converse
2.
23 13 34 21

3.
26

26

x
18

x

698
4.
34

26.4
5.
22

715
x
14

26.7
6.

595
24

24.4
x
42 24

x

16

1640

40.5

60

7.7

135

11.6

Answers

Determine whether

GHI is a right triangle for the given vertices. Explain. 1) 8. G( 6, 2), H(1, 12), I( 2, 1)

7. G(2, 7), H(3, 6), I( 4,

yes; GH

2, HI

98,

no; GH

149, HI 17,
2

130,

IG

(

2)

100,

2

(

98 )

2

(

100 )
4)

IG

2

(

130 )

(

17 )

2

(
9)

149 )

2

(Lesson 7-2)

9. G( 2, 1), H(3,

1), I( 4,

10. G( 2, 4), H(4, 1), I( 1,

yes; GH

29, HI

58,

yes; GH

45, HI

125,

IG

(

29 )

29,

2

(

29 )

2

(

58 )

IG

2

(

45 )

170,

2

(

125 )

2

(

170 )

2

Determine whether each set of measures can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean triple. 11. 9, 40, 41 12. 7, 28, 29 13. 24, 32, 40

...NOTES 5 New Theorems: Parts of Similar Triangles Unit 6 Day 3
1) If two triangles are similar, then the perimeters are proportional to the measures of the corresponding sides. The ratio of the perimeters = scale factor.
2) If two triangles are similar, the measures of the corresponding altitudes are proportional to the measure if the corresponding sides.
3) If the two triangles are similar. The the measures of the corresponding angle bisectors fo the triangles are proportional to the measure of the corresponding sides. Ratio of angle bisectors = scale factor
4) If the two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. Ratio of medians=scale factor
SUMMARY:
If triangle A is similar to triangle B then the ratio of all corresponding one dimensional lengths
is = to the scale factor.
5) Angle bisector proportionality theorem
An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides.
Classwork Parts of Similar Triangles Unit 6 Day 3
Find the value of each variable in each figure below.
...

...+ cos A sin A − cos A 2 + = sin A − cos A sin A + cos A sin 2 A − cos 2 A
AE DE = ? Justify your answer. CE BE
21. Observe the graph given below and state whether triangle ABC is scalene, isosceles or equilateral. Justify your answer. Also find its area.
15. A bag contains 5 red, 8 green and 7 white balls. One ball is drawn at random from the bag, find the probability of getting (i) a white ball or a green ball. (ii) neither a green ball nor a red ball. OR One card is drawn from a well shuffled deck of 52 playing cards. Find the probability of getting (i) a non-face card (ii) A black king or a red queen. Section C 16. Using Euclid’s division algorithm, find the HCF of 56, 96 and 404. OR Prove that 3 − 5 is an irrational number. 17. If two zeroes of the polynomial x4 + 3x3 – 20x2 – 6x + 36 are 2 and − 2 , find the other zeroes of the polynomial. 18. Draw the graph of the following pair of linear equations. x + 3y = 6 2x – 3y = 12 Hence find the area of the region bounded by x = 0, y = 0 and 2x – 3y = 12. 19. A contract on construction job specifies a penalty for
2
22. Find the area of the quadrilateral whose vertices taken in order are A(–5, –3) B(– 4, –6), C(2, –1) and D(1, 2). 23. Construct a DABC in which CA = 6 cm, AB = 5 cm and Ð BAC = 45o, then construct a triangle similar to the given triangle whose sides are
6 of the corre5
sponding sides of the DABC. 24. Prove that the intercept of a tangent between two...

...Oblique Triangles, Laws of Sines and Cosines
INTRODUCTION:
Student will demonstrate how to apply laws of sines and cosines to oblique triangles.
OBJECTIVES:
After completing this unit, the student will be able to:
6. Use the Law of Sines and the Law of Cosines to solve oblique triangle problems.
6.1. Summarize the Law of Sines.
6.2. Find the area of an oblique triangle using the sine function.
6.3. Judge when an ambiguous case of the Law of Sines occurs.
6.4. Solve applied problems using the Law of Sines.
6.5. Summarize the Law of Cosines.
6.6. Use the Law of Cosines to solve oblique triangle problems.
6.7. Solve applied problems using the Law of Cosines.
6.8. Find the area of an oblique triangle using Heron’s formula.
PROCEDURE:
Content
Activity
Objectives
Present objectives and purpose of lesson.
Law of Sines and Law of Cosines
Generalize the sine and cosine relations of the right triangles to oblique triangles by defining the two laws.
Law of Sines: This law relates the three sides of any triangle to the angles opposite the sides, typically labeled a, b, and c for the sides and A, B, and C for the angles.
Law of Cosines: This law relates one side to the other two sides and its corresponding angle:
Relate that either of these relations reduce to simpler forms for the case of right triangles, particularly:
1. Law of Sines...

...PERIOD
FEATURES
REMARKABLE CHANGES
PRE-SPANISH PERIOD
Do not have an organized system of education as we have now.
They followed their code of laws “the Code of Kalantiao and Maragtas.
Ideas and facts were acquired through suggestion, observation, example and imitation.
The youngsters learned by experienced and the learned more in occupational.
The inhabitants were civilized people, possessing their system of writing, laws and moral standards in a well-organized system of government
THE SPANISH-DEVISED CURRICULUM
Three R’s –Reading, Writing and Religion
The curriculum is more on Catholicism and the acceptance of Spanish rule.
The main reading materials were the Cartilla, Caton, and Catecismo.
The curriculum for boys and girls was aimed to teach young boys and girls to serve God, discover what is good and proper for one’s self and enable the individual to get along well with his neighbors
THE AMERICAN-DEVISED CURRICULUM
The public school system established.
Train the Filipinos after the American culture and way of life.
The curriculum is based on the Ideals and traditions of America and her hierarchy values.
Filipinos taught to draw chimneys and play the role of Indian and Cowboys.
Body training – Singing, Drawing, Handwork and PE.
Mental Training – English (reading, writing, conversation, phonetics
and spelling) Nature
study and arithmetic.
In Grade 3...

............................................................................... 4
Calculated….................................................................................. ….................................. 5
Data findings........................................................................................................................ 6
RECOMMENDATIONS and CONCLUSIONS ................................................................ 7
AIM
To show that three forces acting upon a body, in equilibrium, may be represented by a triangle of forces. (Vector addition)
INTRODUCTION
With this experiment we will show that a body in a state of equilibrium, with three forces acting in a singular plane. The following conditions must be met:
* Moment of all three forces must pass through the same point.
* Magnitudes of the forces can be represented by the sides of a triangle, each as a vector with the side of the triangle being the magnitude of the force and the angle being the heading of the force.
EQUIPMENT and COMPONENTS USED
* Vertical fixed force
* String
* A3 drawing paper
* 2x free running pulleys
* Assorted masses
* Straight edge and set square
* Protractor
EXPERIMENTAL METHOD AND PROCEDURE
Part 1
1. Tape the drawing paper to the vertical board and attach pulleys to the edges of the board near the top.
2. Run the string through the two pulleys, hang two masses at...

...
5C Problems involving triangles
cQ1. The diagram shows a sector AOB of a circle of radius 15 cm
and centre O.
The angle at the centre of the circle is 115.
Calculate (a) the area of the sector AOB.
(b) the area of the shaded region. (226 , 124
nQ2. Consider a triangle and two arcs of circles.
The triangle ABC is a right-angled isosceles
triangle, with AB = AC = 2.
The point P is the midpoint of [BC].
The arc BDC is part of a circle with centre A.
The arc BEC is part of a circle with centre P.
(a) Calculate the area of the segment BDCP.
(b) Calculate the area of the shaded region BECD.
cQ3. In the following diagram, O is the centre of the circle
and (AT) is the tangent to the circle at T.
If OA = 12 cm, and the circle has a radius of 6 cm,
ﬁnd the area of the shaded region.
cQ4. The diagram shows a circle, centre O, with a radius 12 cm.
The chord AB subtends at an angle of 75° at the centre.
The tangents to the circle at A and B meet at P.
(a) Using the cosine rule, show that the length of AB
is
(b) Find the length of BP.
(c) Hence find
(i) the area of triangle OBP;
(ii) the area of triangle ABP.
(d) Find the area of sector OAB.
(e) Find the area of the shaded region.
Miscellaneous Problems
Q5. The diagram below...

...irrational
B.
negative and rational
C.
positive and irrational
D.
positive and rational
2]
The value of the polynomial x2 – x – 1 at x = -1 is
[Marks:1]
A.
Zero
B.
-1
C.
-3
D.
1
3]
The remainder when x2 + 2x + 1 is divided by (x + 1) is
[Marks:1]
A.
1
B.
4
C.
-1
D.
0
4]
In fig., AOB is a straight line, the value of x is:
[Marks:1]
A.
60°
B.
20°
C.
40°
D.
30°
5]
The number of line segment determined by three given non - collinear points is:
[Marks:1]
A.
Two
B.
infinitely many
C.
Four
D.
Three
6]
The area of a right triangle with base 5 m and altitude 12 m is
[Marks:1]
A.
50 m2
B.
15 m2
C.
9 m2
D.
30 m2
7]
Evaluate: 53 - 23 - 33
[Marks:1]
A.
80
B.
60
C.
120
D.
90
8]
The area of an equilateral triangle of side 14 cm is
[Marks:1]
A.
B.
C.
D.
9]
Simplify:
[Marks:2]
10]
Check whether (x + 1) is a factor of x3 + x + x2 + 1.
[Marks:2]
11]
In fig., OQ bisects AOB. OP is a ray opposite to ray OQ. Prove that POA=POB.
OR
In fig., AOC and BOC form a linear pair. If a - b = 80°, find the values of a and b.
[Marks:2]
12]
[Marks:2]
13]
The perpendicular distance of a point from the x - axis is 2 units and the perpendicular distance from the y - axis is 5 units. Write the coordinates of such a...

...Problems with Similar Triangles
In the previous document in this series, we defined the concept of similar triangles, ∆ABC ∼ ∆A’B’C’ as a pair of triangles whose sides and angles could be put into correspondence in such a way that it is true that property (i): A = A’ and B = B’ and C = C’. property (ii):
a b c = = a' b' c '
If property (i) is true, property (ii) is guaranteed to be true. If property (ii) is true, then property (i) is guaranteed to be true. We also demonstrated some strategies for establishing that two triangles are similar using property (i). This is very useful to be able to do, since then, we may be able to use the property (ii) conditions to calculate unknown lengths in the triangles. Example 1: Given that lines DE and AB are parallel in the figure to the right, determine the value of x, the distance between points A and D. solution: First, we can demonstrate that ∆CDE ∼ ∆CAB because C=C and ∠CDE = ∠CAB because line AC acts as a transversal across the parallel lines AB and DE, and since ∠CDE and ∠CAB are corresponding angles in this case, they are equal. Since two pairs of corresponding angles are equal for the two triangles, we have demonstrated that they are similar triangles. To avoid error in exploiting the similarity of these triangles, it is useful to redraw them as separate triangles:
B 11
D 7 E
B E 11 A 7 x...

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