2) Similar Triangles Similar triangles have the same shape; corresponding angles are equal and corresponding sides are proportional. 3) Area of a Triangle Area of Triangle = 1/2 (Base) (Height) The height is the perpendicular distance between the side that is chosen as the base and the opposite vertex. Example: If a triangle of base 6 has the same area as a circle of radius 6, what is the altitude of the triangle? Solution: The area of the circle is (6)(6) = 36. In the triangle: 1/2 (6) Height = 36 pi solving for Height, Height = 12 pi 4) Triangle Inequality Theorem The length of one side of a triangle must be greater than the difference and less than the sum of the lengths of the other two sides. Example: if it is given that the length of one side is 3 and the length of another side is 7, then you know that the length of the third side must be greater than 7 -3 = 4 and less than 7 + 3 = 10. 5) Isosceles Triangles An isosceles triangle is a triangle that has two equal sides. Not only are two sides equal, but the angles opposite the equal sides, called base angles, are also equal. Example: The vertex angle of an isosceles triangle is p degrees. How many degrees are there in one of the base angles? Solution: There are (180 - p) degrees left, which must be divided by two congruent angles. Each angle will contain (180 - p) / 2, or 90 - p/2 degrees. 6) Equilateral Triangles In equilateral triangles, all three sides (and all three angles) are equal. All three angles in an equilateral triangle measure 60 degrees, regardless of the lengths of sides. Q. RIGHT TRIANGLES By definition, a right triangle contains a 90 degree angle. 1) Pythagorean Theorem For all right triangles: 2 2 2 (leg) + (1eg) = (hypotenuse) In this case, (2)(2) + (3)(3) = 4 + 9 = 13. Thus, the hypotenuse is the square root of 13. Example: A strobe light is 5 feet from one wall of a room and 10 feet from the wall at right angles to it. How many feet is it from the intersection of the two walls? Solution: The situation is describing a right triangle in which the hypotenuse is the unknown variable. solve by using the Pythagorean theorem: (5)(5) + (10)(10) = xx, x = 5 times the square root of 5 Example: If ABC is a right triangle with a right angle at B, and if AB = 6 and BC = 8, what is the length of AC? 2 2 2 Solution: Use the Pythagorean theoremL AB + BC = AC (6)(6) + (8)(8) = 100 AC = 10 2) The 3-4-5 Triangle If a right triangle's leg-to-leg ratio is 3:4, or if the leg-to-hypotenuse ratio is 3:5 or 4:5, it's a 3-4-5 triangle. In this case, we don't need to use the Pythagorean theorem to find the third side. Just figure out what multiple of 3-4-5 it is: In the right triangle shown, one leg is 30 and the hypotenuse is 50. This is 10 times 3-4-5. We therefore know that the other leg is 40. 3) 5-12-13 Triangle If a right triangle's leg-to-leg ratio is 5:12, or if the leg-to-hypotenuse ratio is 5:13 or 12:13, then it's a 5-12-13 triangle. In this case, we don't need to use the Pythagorean theorem to find the third side. Just figure out what multiple of 5-12-13 it is. Here one leg is 36 and the hypotenuse is 39. This is 3 times 5-12-13. The other leg is 15. Example: What is the area of a right triangle with sides 5, 12 and 13? Solution: The triangle has a hypotenuse of 13 and legs of 12 and 5. Since the legs are perpendicular to each other, we can use one as the base and the other as the height of the triangle. Area =1/2 bh = 1/2 (12)(5) = 30 d) 30-60-90 Triangle __ The sides of a 30-60-90 triangle are in a ratio of x : x V 3 : 2x. We don't need to use the Pythagorean theorem. e) 45-45-90 Triangle __ The sides of a 45-45-90 triangle are in a ratio of x : x : x V 2 . If one leg is 3, then the other leg is also 3, and the hypotenuse is equal to a leg times the square root of two, or 3 times the square root of two. R. OTHER POLYGONS 1) Characteristics of a Rectangle A rectangle is a four-sided figure with four right angles. Opposite sides are equal. Diagonals are equal. B C Quadrilateral ABCD above is shown to have three right angles. The fourth angle therefore also measures 90°, and ABCD is a rectangle. The perimeter of a rectangle is equal to the sum of the lengths of the four sides, which is equivalent to 2(Length + Width). 2) Area of a Rectangle: Area of Rectangle = length X width Example: Find the altitude of a rectangle if its area is 320 and its base is 5 times its altitude. Solution: Let the altitude be b. The base is 5b, and the Area = bh. Area - (5b)(b) = 320 Solving for b, b = the square root of 64 =8 3) Characteristics of a Parallelogram A parallelogram has two pairs of parallel sides. opposite sides are equal. Opposite angles are equal. Consecutive angles add up to 180 degrees. Example: In parallelogram ABCD, angle A is four times angle B. What is the measure in degrees of angle A? Solution: The consecutive angles of a parallelogram are supplementary, so: x + 4x = 180, solving forx, x = 36. Thus, angle A is 4(36) = 144 degrees 4) Area of a Parallelogram: Area of Parallelogram = Base X Height In parallelogram KLMN above, 4 is the height when LM or KN is used as the base. Base X Height = 6 X 4 = 24. Example: If the base of a parallelogram decreases by 20% and the height increases by 40%, by what percent does the area increase? Solution: The area of the original parallelogram = Base X Height. Let b = the length of the base and h = the height of the original parallelogram. If the base decreases by 20%, it becomes .8b. If the height increases by 40%, it becomes 1.4h. The new area is therefore: A = (0.8)b (1.4)h = 1.12 bh, which is 12% bigger than the original area. 5) Characteristics of a Square A square is a rectangle with four equal sides. If PQRS is a square, all sides are the same length as QR. The perimeter of a square is equal to four times the length of one side. 2 6) Area of a Square: Area of Square = (side) In the square above with sides of length 2, the area is 2 x 2 = 4. Example: If the area of a square of side x is 5, what is the area of a square of side 3x? Solution: If the sides have a ratio of 1:3, then theareas have a ratio of 1:9. Therefore, the area of the larger square is 5(9) = 45. Example: Find the area of a square whose diagonal is 12 feet. Solution: Let s = a side of the square. Knowing the the square is actually 2 triangles that share the same hypotenuse (the diagonal), we can use the Pythagorean theorem to solve for the length of a side. (s)(s) + (s)(s) = (12)(12) 2(s)(s) = 144 Side length = square root of 72 S. CIRCLES 1) Characteristics of Circles Circles are closed plane curves with all points on the curve equally distant from a fixed point called the center. A radius of a circle is a line segment from the center to any point on the circle. All radii of a circle are equal. A chord is a line segment whose endpoints are on the circle. A diameter of a circle is a chord that passes through the center of the circle. The diameter of a circle is twice its radius and the longest distance between two points on the circle. An arc is a portion of a circle, usually measured in degrees. The entire circle is 360 degrees A semicircle (half a circle) is 180 degrees A quarter of a circle is an arc of 90 degrees A central angle is an angle whose vertex is the center of the circle and whose sides are radii of the circle. A central angle is equal in measure to its arc. An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle. An inscribed angle is equal in measure to one-half its arc. 2) Circumference of a Circle: 2 times pi times the radius In the circle above, the radius is 3, and so the circumference is 2 x pi x 3 = 6 pi. 3) Length of an Arc An arc is a piece of the circumference. If n is the degree measure of the arc's central angle, then the formula is: Length of an Arc = (n/360)(2 x pi x r) In the figure above, the radius is 5 and the measure of the central angle is 72°. The arc length is 72/360 or 1/5 of the circumference: (72/360) ( 2 x pi) (5) = (1/5) (10 x pi) = 2 pi Example: If a circle of radius 3 feet has a central angle of 60 degrees, find the length of an arc intercepted by this central angle. Solution: Arc =(60/360) (2)(3) pi = pi feet 4) Area of a Circle: Area of a Circle = 2 x pi x pi or 2 pi The area of the circle shown is (4)(4) pi = 16 pi. Example: What is the area of the circle that passes through the point (10, 8) and has its center at (2, 2)? Solution: We can use the distance formula to dind the radius of the circle: Radius = Square root of { (10- 2)(10 - 2) + (8 - 2)(8 - 2) } = 10 Thus, the radius of the circle is 10. The Area of the circle = (10)(10) pi = 100 pi Example: If the radius of a circle is decreased by 10%, by what percent is its area decreased? Solution: If the radii of the two circles have a ratio of 10:9, the areas have a ratio of 100:81. Therefre, the decrease is is 19 out of 100, or 19% 5) Area of a Sector A sector is a piece of the area of a circle. If n is the degree measure of the sector's central angle, then the formula is: Area of a Sector = (n/360)(pi)(r)(r) In the figure above, the radius is 6 and the measure of the sector's central angle is 30°, The sector has 30/360 or 1/12 of the area of the circle: (30/360)(pi)(6)(6) = (1/12)(36)(pi) = 3 pi T. SOLIDS 1) Surface Area of a Rectangular Solid The surface of a rectangular solid consists of three pairs of identical faces. To find the surface area, add the area of each face. If the length is l, the width is w, and the height is h, the formula is: Surface Area = 2lw + 2wh + 2lh The rectangular solid shown above has a length of 7, a width of 3 and a height of 4. It's surface area is 2(7)(3) + 2 (3)(4) + 2 (7)(4) = 42 + 24 + 56 = 122 2) Volume of a Rectangular Solid = Length x Width x Height The volume of a 4-by-5-by-6 box is 4 x 5 x 6 = 120 A cube is a rectangular solid with length, width, and height all equal. If a is the length of an edge of a cube, the volume formula is: (a)(a)(a) For a cube with a side length of 2, the volume is (2)(2)(2) = 8. Example: If the surface area of a cube is 150 square feet, how many cubic feet are there in the volume of the cube? Solution: The surface area of the cube is composed of 6 equal sides. If each edge of the cube is x, then 6xx = 125. Solving for x, x = 5. The volume is x cubed, or (5)(5)(5) = 125. Example: What is the surface area of a cube whose volume is 125 cubic centimeters? Solution: The volume = (s)(s)(s), where s is the length of a side. Thus, a side is the cubic root of 125, or 5 centimeters. If the side of the cube is 5 cm, the area of one of its faces is (5)(5) = 25 square centimeters. Since a cube has 6 faces, its surface area is 6 x 25 = 150 square centimeters. 2 3) Volume of a Cylinder = pi x r x Height In the cylinder above, r = 2 and h = 5, so Volume = pi (2)(2)(5) = 20 pi Example: A cylindrical pail has a radius of 7 inches and a height of 9 inches. If there are 231 cubic inches to a gallon, approximately how many gallons will this pail hold? Solution: Use the formula Volume = pi (r)(r) h = 3.1416(7)(7)(9)(9) = 8.2 gallons Example: The volume of a cylinder having a height of 12 is 144 pi. What is the radius of its base? Solution: The formula for the volume of a cylinder is V = pi (r)(r)h. Solving for r, we get r = 2 times the square root of 3. . than the difference and less than the sum of the lengths of the other two sides. Example: if it is given that the length of one side is 3 and the length of another side is 7, then you know that the. central angle is 30 °, The sector has 30 /36 0 or 1/12 of the area of the circle: (30 /36 0)(pi)(6)(6) = (1/12) (36 )(pi) = 3 pi T. SOLIDS 1) Surface Area of a Rectangular Solid The surface of a rectangular. circumference. If n is the degree measure of the arc's central angle, then the formula is: Length of an Arc = (n /36 0)(2 x pi x r) In the figure above, the radius is 5 and the measure of the central