Quadrilaterals A quadrilateral is a four-sided polygon. Since a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360 degrees. m∠1 + m∠2 + m∠3 + m∠4 + m∠5 + m∠6 = 360° Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. In the figure above, AB || CD and BC || AD . Parallel lines are symbolized with matching numbers of trian- gles or arrows. A parallelogram has: ■ opposite sides that are congruent (A ෆ B ෆ = C ෆ D ෆ and B ෆ C ෆ = A ෆ D ෆ ) ■ opposite angles that are congruent (m∠A = m∠C and m∠B = m∠D) ■ consecutive angles that are supplementary (m∠A + m∠B = 180°, m∠B + m∠C = 180°, m∠C + m∠D = 180°, m∠D + m∠A = 180° ) ■ diagonals (line segments joining opposite vertices) that bisect each other (divide each other in half) SPECIAL TYPES OF PARALLELOGRAMS ■ A rectangle is a parallelogram that has four right angles. y x x y A B C D 1 3 4 5 6 2 – THEA MATH REVIEW– 165 ■ A rhombus is a parallelogram that has four equal sides. ■ A square is a parallelogram in which all angles are equal to 90 degrees and all sides are congruent. A square is a special case of a rectangle where all the sides are congruent. A square is also a special type of rhombus where all the angles are congruent. D IAGONALS OF PARALLELOGRAMS In this diagram, parallelogram ABCD has diagonals AC and BD that intersect at point E. The diagonals of a par- allelogram bisect each other, which means that AE ≅ EC and BE ≅ ED . In addition, the following properties hold true: ■ The diagonals of a rhombus are perpendicular. ■ The diagonals of a rectangle are congruent. ■ The diagonals of a square are both perpendicular and congruent. Example In parallelogram ABCD, the diagonal AC = 5x + 10 and BC = 9x. Determine the value of x. C A B D E A B D C Rectangular ABCD AC ≅ BD Rhombus ABCD AC Ќ BD Square ABCD AB Ќ CD, AC ≅ BD A B D C x x x x x x x x – THEA MATH REVIEW– 166 Since the diagonals of a parallelogram are congruent, the lengths are equal. We can then set up and solve the equation 5x + 10 = 9x to determine the value of x. 5x + 10 = 9x Subtract x from both sides of the equation. 10=4x Divide 4 from both sides of the equation. 2.5 = x AREA AND VOLUME FORMULAS The area of any parallelogram can be found with the formula A = bh,where b is the base of the parallelogram, and h is the height. The base and height of a parallelogram is defined the same as in a triangle. Note: Sometimes b is called the length (l) and h is called the width (w) instead. If this is the case, the area formula is A = lw. A rectangular prism (or rectangular solid) is a prism that has rectangles as bases. Recall that the formula for any prism is V = A b h. Since the area of the rectangular base is A = lw, we can replace lw for A b in the formula giving us the more common, easier to remember formula, V = lwh. If a prism has a different quadrilateral-shaped base, use the general prisms formula for volume. Note: A cube is a special rectangular prism with six congruent squares as sides. This means that you can use the V = lwh formula for it, too. Rectangular Solid h l w V = lwh b h b h h b h b – THEA MATH REVIEW– 167 Circles TERMINOLOGY A circle is formally defined as the set of points a fixed distance from a point. The more sides a polygon has, the more it looks like a circle. If you consider a polygon with 5,000 small sides, it will look like a circle, but a circle is not a polygon. A circle contains 360 degrees around a center point. ■ The midpoint of a circle is called the center. ■ The distance around a circle (called perimeter in polygons) is called the circumference. ■ A line segment that goes through a circle, with its endpoints on the circle, is called a chord. ■ A chord that goes directly through the center of a circle (the longest line segment that can be drawn) in a circle is called the diameter. ■ The line from the center of a circle to a point on the circle (half of the diameter) is called the radius. ■ A sector of a circle is a fraction of the circle’s area. ■ An arc of a circle is a fraction of the circle’s circumference. CIRCUMFERENCE, AREA, AND VOLUME FORMULAS The area of a circle is A = πr 2 ,where r is the radius of the circle. The circumference (perimeter of a circle) is 2πr, or πd,where r is the radius of the circle and d is the diameter. Example Determine the area and circumference of the circle below: 6' Center A Radius AB Diameter EF Chord PN A B F A E O P N – THEA MATH REVIEW– 168 We are given the diameter of the circle, so we can use the formula C = πd to find the circumference. C = πd C = π(6) C = 6π ÷ 18.85 feet The area formula uses the radius, so we need to divide the length of the diameter by 2 to get the length of the radius: 6 ÷ 2 = 3. Then we can just use the formula. A = π(3)2 A = 9π ÷ 28.27 square feet. Note: Circumference is a measure of length, so the answer is measured in units, where the area is measured in square units. AREA OF SECTORS AND LENGTHS OF ARCS The area of a sector can be determined by figuring out what the percentage of the total area the sector is, and then multiplying by the area of the circle. The length of an arc can be determined by figuring out what the percentage of the total circumference of the arc is, and then multiplying by the circumference of the circle. Example Determine the area of the shaded sector and the length of the arc AB. Since the angle in the sector is 30°, and we know that a circle contains a total of 360°, we can deter- mine what fraction of the circle’s area it is: ᎏ 3 3 6 0 0 ° ° ᎏ = ᎏ 1 1 2 ᎏ of the circle. The area of the entire circle is A = πr 2 , so A = π(4) 2 = 16π. So, the area of the sector is ᎏ 1 1 2 ᎏ ϫ 16π = ᎏ 1 1 6 2 π ᎏ = ᎏ 4 3 ᎏ π≈4.19 square inches. We can also determine the length of the arc AB, because it is ᎏ 3 3 6 0 0 ° ° ᎏ = ᎏ 1 1 2 ᎏ of the circle’s circumference. The circumference of the entire circle is C = 2πr, so C = 2π(4) = 8π. This means that the length of the arc is ᎏ 1 1 2 ᎏ ϫ 8π = ᎏ 8 1 π 2 ᎏ = ᎏ 3 2 π ᎏ ≈ 2.09 inches. 4' B A 30˚ – THEA MATH REVIEW– 169 A prism that has circles as bases is called a cylinder. Recall that the formula for any prism is V = A b h. Since the area of the circular base is A = πr 2 , we can replace πr 2 for A b in the formula, giving us V = πr 2 h,where r is the radius of the circular base, and h is the height of the cylinder. A sphere is a three-dimensional object that has no sides. A basketball is a good example of a sphere. The vol- ume of a sphere is given by the formula V = ᎏ 4 3 ᎏ πr 3 . Example Determine the volume of a sphere whose radius is 1.5'. Replace 1.5' in for r in the formula V = ᎏ 4 3 ᎏ πr 3 . V = ᎏ 4 3 ᎏ πr 3 V = ᎏ 4 3 ᎏ π(1.5) 3 V = ᎏ 4 3 ᎏ (3.375)π V = 4.5π≈14.14 The answer is approximately 14.14 cubic feet. Example An aluminum can is 6" tall and has a base with a radius of 2". Determine the volume the can holds. Aluminum cans are cylindrical in shape, so replace 2" for r and 6" for h in the formula V = πr 2 h. V = πr 2 h V = π(2) 2 (6) V = 24π≈75.40 cubic feet Cylinder V = πr 2 h r h – THEA MATH REVIEW– 170 . into two triangles, the sum of its interior angles will equal 180 + 180 = 360 degrees. m∠1 + m∠2 + m∠3 + m∠4 + m∠5 + m 6 = 360 ° Parallelograms A parallelogram is a quadrilateral with two pairs of. circumference. C = πd C = π (6) C = 6 ÷ 18.85 feet The area formula uses the radius, so we need to divide the length of the diameter by 2 to get the length of the radius: 6 ÷ 2 = 3. Then we can just. total of 360 °, we can deter- mine what fraction of the circle’s area it is: ᎏ 3 3 6 0 0 ° ° ᎏ = ᎏ 1 1 2 ᎏ of the circle. The area of the entire circle is A = πr 2 , so A = π(4) 2 = 16 . So, the