Geometry: Trimester 2 Final Review

• Summary of Trimester 2 Final Review, covering geometry problems.
• Includes side length calculations, triangle classifications, trigonometric functions, reflections, rotations, translations, and circle properties.
• Also includes area and volume calculations for 2D and 3D shapes.

Missing Side Lengths

• For a right triangle with legs of 14 inches and 3 inches, the hypotenuse is approximately 14.32 inches.
• In a right triangle, if one leg is 6 ft and the hypotenuse is 9 ft, the other leg is approximately 6.71 ft.
• For a right triangle with one leg 4 in and hypotenuse 5 in, the other leg is 3 in.

Triangle Classification

• Sides 5, 12, and 13 form a right triangle because 5²+12²=13².
• Sides 3, 5, and 7 form an obtuse triangle because 3²+5²<7².
• Sides 7, 8, and 9 form an acute triangle because 7²+8²>9².

Triangle Measurements

• Side BT in the triangle is 8.
• Side AT is 8√2.
• Side OB is 10/√2.
• Side BY is 10/√2.

Missing Side Lengths (Radical Form)

• In a 30-60-90 triangle with hypotenuse 20, the side opposite the 60° angle is 10√3, and the side opposite the 30° angle is 10.
• The side opposite the 30° angle is 5√3, so the side is 5.
• For a 45-45-90 triangle with a leg of 6√2, the hypotenuse is 12.
• For a 30-60-90 triangle with the side opposite the 60° angle equal to √15, the side opposite the 30° angle is √5.

Trigonometric Ratios

• sinA = 5/13, cosA = 12/13, tanA = 5/12
• sinB = 12/13, cosB = 5/13, tanB = 12/5

Solving for Missing Side Lengths (with Trigonometry)

• Given cos(29°) = 7/x, x ≈ 7
• Given sin(29°) = y/7, y ≈ 3.39
• Given tan(48°) = x/5, x ≈ 5.55
• Given cos(48°) = 5/y, y ≈ 7.47

Solving for Missing Angles (with Trigonometry)

• Given sin(x) = 8/11, x ≈ 46.67°
• Given tan(y) = 31/23, y ≈ 53.43°

Reflections

Reflection over y=x

• A' = (0, -4) becomes A'(0,-4), B'(-4, 0) becomes B'(-4,0), , and C'(-4, -4) becomes C'(-4,-4).

Reflection over y-axis

• A'(-4, 0) becomes A'(4,0), B'(0, -4) becomes B'(0,4), and, C'(-4, -4) becomes C'(4,4).

Reflection over x-axis

• A'(-4, 0) becomes A'(-4,0), B'(0, -4) becomes B'(0,4), and C'(-4, -4) becomes C'(-4,4).

Rotations (Clockwise)

90 degrees

• A(-4, 0) → A'(0, 4)
• B(0, -4) → B'(-4, 0)
• C(-4, -4) → C'(-4, -4)

270 degrees

• A(-4, 0) → A (0, 4)
• B(0, -4) → B'(-4, 0)
• C(-4, -4) → C'(-4, -4)

180 degrees

A(-4, 0) →A'(4, 0)

• B(0, -4) → B'(0, 4)
• C(-4, -4) → C'(4, 4)

Translations

• For the translation function f(x, y) = (x + 3, y - 2):
• A(-4, 0) becomes A'(-1, -2)
• B(0, -4) becomes B'(3, -6)
• C(-4, -4) becomes C'(-1, -6)

Circle Properties

• If the diameter is 8 cm, the circumference is approximately 25.13 cm.
• If the radius is 5 cm, the circumference is approximately 31.42 cm.
• A circle with a circumference of 65.97 inches has a diameter of approximately 21 inches and a radius of approximately 10.5 inches.
• Measures of arcs on circle C: arc AE is 140°, arc BD is 50°, arc ABD is 180°, arc DAB is 220°.
• Arc lengths on circle R:
  • Radius is 7 ft, arc DC is approximately 12.22 ft.
  • Diameter is 14 ft, arc EA is approximately 15.39 ft.
  • Radius is 10 inches, arc DA is approximately 13.96 inches, and arc CB is approximately 8.72 inches.
• Angle D is 38 degrees.
• Arc DC is 90.
• Given inscribed angles:
  • The value of x is 4.
  • The measure of angle A is 70°.
  • The value of y is 10.
  • The measure of angle B is 47°.
• Given the circle equation (x - 3)² + (y + 6)² = 25:
  • The radius is 5.
  • The circumference is approximately 31.42.
  • The center is (3, -6).

Area of Figures

• For a parallelogram with base 15 and height ~15.79, the area is ~192.64.
• For a triangle with base 21 mm and height 9 mm, the area is ~189 mm².
• The area of triangle = 114
• Area of circle = 3421.19
• The area of a shape composed of a rectangle and half circles is 69.82.
• Area of octagon is equal to 371.91.
• Area value shown is 1269.5.

3D Shapes

• Hemisphere: Surface area ~ 190.85. with r = 4.5
• Square Prism: SA= 544, V = 2304
• Cylinder Total SA = 119.37, Volume = 56.55
• Area values shown are 56, and 317.33.
• Area values shown are 50.27, and 150.81.

Comprehensive review of geometry concepts for Trimester 2 finals. Covers calculations of side lengths, triangle classifications based on sides. It also tests understanding of trigonometric functions, transformations, and circle properties, in addition to area and volume for 2D/3D shapes.