Recomendaciones

1 qt 5 .95 l 1 gal 5 4 qts

1 gal 5 4 3 .95 l 5 3.8 l

Multiply to find the number of liters in 5 gallons:

5 gal 5 5 3 3.8 l 5 19 l 19. The correct answer is (B).

°F 59

5312° 1 32°

5108°

5 132°

521.6° 1 32°

553.6°

So the correct answer is 54°.

20. The correct answer is (C).

°C 55

9~68° 2 32°! 55

9 3 36°

1 4

GEOMETRY

ANGLES

1. a. An angle is the figure formed by two lines meeting at a point.

b. The point B is the vertex of the angle and the lines BA and BC are the sides of the angle.

2. There are three common ways of naming an angle:

a. By a small letter or figure written within the angle as∠m.

b. By the capital letter at its vertex, as∠B.

c. By three capital letters, the middle letter being the vertex letter, as ∠ABC or

∠CBA.

3. a. When two straight lines intersect (cut each other), four angles are formed. If these four angles are equal, each angle is called a right angle and contains 90°.

The symbol ¡ is used to indicate a right angle.

Example:

∠ABC is a right angle.

b. An angle measuring less than 90° is called an acute angle.

c. If the two sides of an angle extend in opposite directions forming a straight line, the angle is called a straight angle and measures 180°.

d. An angle measuring more than 90° but less than 180° is called an obtuse angle.

4. a. Two angles are complementary if the sum of their measures is 90°.

b. To find the complement of an angle, subtract the measure of the given angle from 90°.

Example: The complement of 60° is 90° 2 60° 5 30°

5. a. Two angles are supplementary if the sum of their measures is 180°.

b. To find the supplement of an angle, subtract the measure of the given angle from 180°.

Example: The supplement of 60° is 180° 2 60° 5 120°.

LINES

6. a. Two lines are perpendicular to each other if they meet to form a right angle.

The symbol ⊥ is used to indicate that the lines are perpendicular.

Example: ∠ABC is a right angle. Therefore AB ⊥ BC.

b. Lines that do not meet no matter how far they are extended are called parallel lines. The symbol \ is used to indicate that the two lines are parallel.

Example: AB \ CD.

A________________B C________________D TRIANGLES

7. A triangle is a closed, three-sided figure. The figures below are all triangles.

8. a. The sum of the measures of three angles of a triangle is 180°.

b. To find the measure of an angle of a triangle when you are given the measures of the other two angles, add the measures of the given angles and subtract their sum from 180°.

Problem: The measures of two angles of a triangle are 60° and 40°. Find the measure of the third angle.

SOLUTION: 60° 1 40° 5 100°

180° 2 100° 5 80°

Answer: The measure of the third angle is 80°.

9. a. A triangle that has two congruent sides is called an isosceles triangle.

b. In an isosceles triangle, the angles opposite the congruent sides are also congruent.

10. a. A triangle that has all three sides congruent is called an equilateral triangle.

b. Each angle of an equilateral triangle measures 60°.

11. a. A triangle that has a right angle is called a right triangle.

b. In a right triangle, the two acute angles are complementary.

c. In a right triangle, the side opposite the right angle is called the hypotenuse and is the longest side. The other two sides are called legs.

Example: ACis the hypotenuse.

ABand BC are the legs.

12. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

13. To find the hypotenuse of a right triangle when given the legs:

a. Square each leg.

b. Add the squares.

c. Compute the square root of this sum.

Problem: In a right triangle the legs are 6 inches and 8 inches. Find the hypotenuse.

SOLUTION: 6²536 8²564 36 1 64 5 100

=

^{100 5 10}

Answer: The hypotenuse is 10 inches.

14. To find a leg when given the other leg and the hypotenuse of a right triangle:

a. Square the hypotenuse and the given leg.

b. Subtract the square of the leg from the square of the hypotenuse.

c. Compute the square root of the difference.

Problem: One leg of a right triangle is 12 feet and the hypotenuse is 20 feet. Find the other leg.

SOLUTION: 12²5144 20²5400 400 2 144 5 256

=

256 5 16 Answer: The other leg is 16 feet.

QUADRILATERALS

15. a. A quadrilateral is a closed, four-sided figure in two dimensions. Common quadrilaterals are the parallelogram, rectangle, and square.

b. The sum of the measures of the four angles of a quadrilateral is 360°.

16. a. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

b. Opposite sides of a parallelogram are congruent.

c. Opposite angles of a parallelogram are congruent.

In parallelogram ABCD,

AB \ CD, AD \ BC AB 5 CD, AD 5 BC m∠A 5 m∠C, m∠B 5 m∠D

17. A rectangle has all of the properties of a parallelogram. In addition, all four of its angles are right angles.

18. A square is a rectangle with the additional property that all four of its sides are congruent.

CIRCLES

19. A circle is a closed-plane curve, all points of which are equidistant from a point within called the center.

20. a. A complete circle contains 360°.

b. A semicircle contains 180°.

21. a. A chord is a line segment connecting any two points on the circle.

b. A radius of a circle is a line segment connecting the center with any point on the circle.

c. A diameter is a chord passing through the center of the circle.

d. A secant is a chord extended outside of the circle in either one or both directions.

e. A tangent is a line touching a circle at one and only one point.

f. The circumference is the length of the curved line bounding the circle.

g. An arc of a circle is any part of the circumference.

22. a. A central angle, as∠AOB in the figure below, is an angle whose vertex is the center of the circle and whose sides are radii. A central angle is equal to, or has the same number of degrees as, its intercepted arc.

b. An inscribed angle, as∠MNP, is an angle whose vertex is on the circle and whose sides are chords. An inscribed angle has half the number of degrees as its intercepted arc. ∠MNP intercepts arc MP and has half the degrees of arc MP.

PERIMETER

23. The perimeter of a two-dimensional figure is the distance around the figure.

Example: The perimeter of the figure above is 9 1 8 1 4 1 3 1 5 5 29.

24. a. The perimeter of a triangle is found by adding the length of all of its sides.

Example: If the sides of a triangle are 4, 5, and 7, its perimeter is 4 1 5 1 7 5 16.

b. If the perimeter and two sides of a triangle are given, the third side is found by adding the two given sides and subtracting this sum from the perimeter.

Problem: Two sides of a triangle are 12 and 15, and the perimeter is 37.

Find the other side.

SOLUTION: 12 1 15 5 27 37 2 27 5 10

Answer: The length of the third side is 10.

25. The perimeter of a rectangle equals twice the sum of the length and the width. The formula is P 5 2(l 1 w).

Example: The perimeter of a rectangle whose length is 7 feet and width is 3 feet equals 2 3 10 5 20 feet.

26. The perimeter of a square equals the length of its side multiplied by 4. The formula is P 5 4s.

Example: The perimeter of a square, one side of which is 5 feet, is 4 3 5 feet 5 20 feet.

27. a. The circumference of a circle is equal to the product of the diameter multiplied by p. The formula is C 5 pd.

b. The number p (“pi”) is approximately22

7, or 3.14 (3.1416 for greater accuracy).

A problem will usually state which approximation for p to use; otherwise, express the answer in terms of “pi,” p.

Example: The circumference of a circle whose diameter is 4 inches is 4p inches; or, if it is approximated by 22

7, then the circumference would be 4 322

7 5 88 7 5 124

7inches.

c. Since the diameter is twice the radius, the circumference equals twice the radius multiplied by p. The formula is C 5 2pr.

Example: If the radius of a circle is 3 inches, then the circumference is 6p inches.

d. The diameter of a circle equals the circumference divided by p.

Example: If the circumference of a circle is 11 inches, then assuming p 5 22

7

diameter 5 11 422 7 inches 1

511 3 7 22inches

2 57

2inches, or 31 2inches.

AREA

28. a. In a figure of two dimensions, the total space within the figure is called the area.

b. Area is expressed in square units, such as square inches, square centimeters, and square miles.

c. In computing area, all dimensions must be expressed in the same units.

29. The area of a square is equal to the square of the length of its side. The formula is A 5 s².

Example: The area of a square, one side of which is 6 inches, is 6 3 6 5 36 square inches.

30. a. The area of a rectangle equals the product of the length and the width. The length is any side; the width is a side next to the length. The formula is A 5 l 3 w.

Example: If the length of a rectangle is 6 feet and its width 4 feet, then the area is 6 3 4 5 24 square feet.

b. If given the area of a rectangle and one dimension, divide the area by the given dimension to find the other dimension.

Example: If the area of a rectangle is 48 square feet and one dimension is 4 feet, then the other dimension is 48 4 4 5 12 feet.

31. a. The altitude, or height, of a parallelogram is a line drawn from a vertex perpendicular to the opposite side or base.

Example: DEis the height.

ABis the base.

b. The area of a parallelogram is equal to the product of its base and its height:

A 5 b 3 h.

Example: If the base of a parallelogram is 10 centimeters and its height is 5 centimeters, its area is 5 3 10 5 50 square centimeters.

c. If given one of these dimensions and the area, divide the area by the given dimension to find the other dimension of a parallelogram.

Example: If the area of a parallelogram is 40 square inches and its height is 8 inches, its base is 40 4 8 5 5 inches.

32. a. The altitude, or height, of a triangle is a line drawn from a vertex perpendicular to the line containing the opposite side, called the base.

b. The area of a triangle is equal to one half the product of the base and the height:

A 51 2b 3 h.

Example: The area of a triangle having a height of 5 inches and a base of 4 inches is1

23 5 3 4 51

2 320 5 10 square inches.

c. In a right triangle, one leg may be considered the height and the other leg the base. Therefore, the area of a right triangle is equal to one half the product of the legs.

Example: The legs of a right triangle are 3 and 4. Its area is1

233 3 4 5 6 square units.

33. a. The area of a circle is equal to the radius squared, multiplied by p: A 5 pr². Example: If the radius of a circle is 6 inches, then the area 5 36p square

inches.

b. To find the radius of a circle given the area, divide the area by p and find the square root of the quotient.

Example: To find the radius of a circle of area 100p, 100p

p 5100

=

100 5 10 5 radius.

34. Some figures are composed of several geometric shapes. To find the area of such a figure it is necessary to find the area of each of its parts.

Problem: Find the area of the figure below.

SOLUTION: The figure is composed of three parts: a square of side 4, a semicircle of diameter 4 (attached to the lower side of the square), and a right triangle with legs 3 and 4 (attached to the right side of the square).

Area of square 5 4²516 Area of triangle 51

433 3 4 5 6 Area of semicircle is1

2area of the circle 51 2pr² Radius 51

234 5 2 Area 51

2pr² 51

23 p 32² 52p

Answer: Total area 5 16 1 6 1 2p 5 22 1 2p

In document PLAN ESTRATÉGICO CAMPOSOL (página 73-81)