Geometry 2011-12-09 www.njctl.org

Pythagorean Theorem Distance Formula Quadrilaterals Midpoint Geometry Pythagorean Theorem Distance Formula Quadrilaterals Midpoint Area and Volume

Pythagorean Theorem Return to the Table of Contents

Recall... right triangle is a triangle with a right (90o) angle. leg hypotenuse The sides form that right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side.

Pythagorean Theorem (R1) In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. leg2 + leg2 = hypotenuse2 or a2 + b2 = c2 a b c

Is the missing side a leg or the hypotenuse of the right triangle? Example Find the length of the missing side of the right triangle. x 9 12 Is the missing side a leg or the hypotenuse of the right triangle? hypotenuse

x 9 12 92 + 122 = x2 81 + 144 = x2 225 = x2 15 = x -15 is a extraneous solution, a distance can not equal a negative number. x = 15 ±

Is the missing side a leg or the hypotenuse of the Example Find the length of the missing side. x 28 20 Is the missing side a leg or the hypotenuse of the right triangle? leg

x 28 20 x2 + 202 = 282 x2 + 400 = 784 x2 = 384 x = 19.60

The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder. Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall will a the ladder reach? 28 feet 7 feet ?

The ladder will reach 18.3 feet up the wall safely. ? a2 + b2 = c2 72 + b2 = 282 49 + b2 = 384 b2 = 335 b 18.30 The ladder will reach 18.3 feet up the wall safely. »

Try this... The dimensions of a high school basketball court are 84' long and 50' wide. What is the length of from one corner of the court to the opposite corner? 84 50 x 842 + 502 = x2 9556 = x2 97.75 = x The court is 97.75 feet Answer

Pythagorean Theorem Applications The Pythagorean Theorem can also be used in figures that contain right angles.

Find the area of the triangle. Example Find the area of the triangle. 13 feet 10 feet A = bh 1 2 The base of the triangle is given, but we need to find the height of the triangle.

By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. 13 feet 5 feet h 52 + h2 = 132 25 + h2 = 169 h2 = 144 h = 12 A = (10)(12) A = (120) A = 60 feet 1 2

Find the perimeter of the rectangle. Try this... Find the perimeter of the rectangle. 8 in 10 in Prect = 2l + 2w x2 + 82 = 102 x = 6 Prect = 2(6) + 2(8) P = 28 inches ANSWER

Converse of the Pythagorean Theorem (R2) If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. a b c A B C If c2 = a2 + b2, then ABC is a right triangle.

Remember c is the longest side Example Tell whether the triangle is a right triangle. Remember c is the longest side D E F 7 24 25 c2 = a2 + b2 252 = 72 + 242 625 = 49 + 576 625 = 625 DEF is a right triangle.

The missing side is the ________ of the right triangle. 1 The missing side is the ________ of the right triangle. A leg 6 9 x B hypotenuse

Find the length of the missing side. 2 Find the length of the missing side. 6 9 x

The missing side is the _________ of the right triangle. 3 The missing side is the _________ of the right triangle. A leg x 15 36 B hypotenuse

Find the length of the missing side. 4 Find the length of the missing side. x 15 36

5 A NBA court is 50 feet wide and the length from one corner of the court to the opposite corner is 106.5 feet. How long is the court? (Round the answer to the nearest whole number) A 94.03 feet B 117.7 feet C 118 feet D 94 feet

Find the area of the rectangle. 6 Find the area of the rectangle. A 120 feet B 84 feet 8 feet 17 feet C 46 inches D 46 feet

Find the area of the triangle. 7 Find the area of the triangle. 7 inches 24 inches

Distance Formula Return to Table of Contents

Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles. Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle.

Relationship between the Pythagorean Theorem & Distance Formula (x1, y1) (x2, y1) (x2, y2) The distance formula calculates the distance using point's coordinates. c Relationship between the Pythagorean Theorem & Distance Formula b a The Pythagorean Theorem states a relationship among the sides of a right triangle. c2= a2 + b2 The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side.

Distance The Distance Formula (x1, y1) (x2, y2) The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula. The Distance Formula The distance 'd' between any two points with coordinates and is given by the formula: (x1, y1) (x2, y2) d = Note: recall that all coordinates are (x-coordinate, y-coordinate).

Example (x1, y1) (x2, y2) Calculate the distance from Point K to Point I (x1, y1) (x2, y2) d = Plug the coordinates into the distance formula Label the points - it does not matter which one you label point 1 and point 2. Your answer will be the same. KI = = = 4.12

Calculate the distance from Point J to Point K Calculate the distance from H to K

Distance Application - Perimeter The vertices of a triangle are (3, -1), (-2, 4) and (5, 6). Find the perimeter.

Distance Application - Circles A circle is centered center at (3, 4) and has a radius of 3 units. Is (5, 2) on, inside of, or outside of the circle?

8 Calculate the distance from Point G to Point K A 6.32 10.95 7.28 7.62 B C D

9 Calculate the distance from Point I to Point H A 4.24 9 3.46 B C D

10 Calculate the distance from Point G to Point H A 16 4.12 2.24 16.03 B C D

Quadrilaterals Return to Table of Contents

Parallelogram Rhombus opposite sides parallel opposites sides parallel opposites sides congruent all sides congruent

Parallelogram Rectangle opposite sides parallel opposites sides parallel opposites sides congruent opposite sides congruent sides are perpendicular (right angles)

opposite sides parallel one pair of opposite sides parallel Square Trapezoid opposite sides parallel one pair of opposite sides parallel all sides congruent sides are perpendicular (right angles)

The vertices of a quadrilateral are A (-2, 4), B (5, 6), C (12, 4) and D (5, 2). Is this a rhombus that is not a square, a rectangle that is not a square, a square, or a trapezoid? B A C D All sides are congruent (equal length), but the sides are not perpendicular. Therefore, quadrilateral ABCD is a rhombus.

The vertices of a quadrilateral are A (-3, 4), B (4, 4), C (6, -3) and D (-5, -3). Is this a rhombus that is not a square, a rectangle that is not a square, a square, or a trapezoid? A B C D One pair of opposite sides is parallel. Therefore, quadrilateral ABCD is a trapezoid.

11 What type of quadrilateral has vertices (5, 4), (11, 8), (6, 9), and (10, 3)? A trapezoid rectangle that is not a square rhombus that is not a square square B C D

12 What type of quadrilateral has vertices (4, 7), (3, 1), (-3, 2), and (-2, 8)? A parallelogram that is not a rectangle/rhombus/square rectangle that is not a square rhombus that is not a square square B C D

Midpoint Return to Table of Contents

The midpoint of a segment is the halfway point between the endpoints. 8 12 10 midpoint

Find the midpoint of the segment. 1 + 4 = 5 = 2.5 2 2

The midpoint formula (in coordinate plane) finds the average of the x-coordinates and the average of y-coordinates of the endpoints. A B 1 2 3 4 5 6 7 8 9 10 10 5 A (5, 3) B (9, 5)

Find the midpoint of the segment with endpoints A (5,12) and B (-4, 8). 5 + -4 , 12 + 8 2 2 1 , 20 2 2 (1/2, 10) ANSWER

10 Endpoint (2, 6) Midpoint (5, 4) Endpoint ( , ) 8 2 Find the missing endpoint C given the midpoint B and the other endpoint A. 10 5 Endpoint (2, 6) Midpoint (5, 4) Endpoint ( , ) -2 +3 8 2 A B 1 2 3 4 5 6 7 8 9 10

Endpoint (-1, 4) Midpoint (3, -2) Endpoint ( , ) 7 -8 The midpoint of a line is (3, -2). If one endpoint is (-1, 4), find the other endpoint. Endpoint (-1, 4) Midpoint (3, -2) Endpoint ( , ) +4 -6 7 -8

13 Find the midpoint of the segment with endpoints (5, 2) and (-1, 7).

14 Find the midpoint of the segment with endpoints (-3, -4) and (-2, 2).

15 The midpoint of a line segment is (-3, 9) and one endpoint is (-1, 6). Find the other endpoint.

16 The midpoint of a line segment is (0, 2) and one endpoint is (-6, 1). Find the other endpoint.

Area and Volume Return to Table of Contents

Area r d { Circle Rectangle Triangle Trapezoid b h b h b1 b2 h

Find the area of the following figures. Circle Rectangle 2 in 6 in 4 cm A = 42 = 50.27 cm2 A = 2*6 = 12 in2 Triangle Trapezoid 15 m 12 m 2 m 3 ft 2 ft A = 1/2 * 2 * 3 = 3 ft2 A = 1/2*(12 + 15)*2 = 27 m2

The bases of a trapezoid are 3x + 4 and 2x - 3, and the height is 4 inches. Write an expression representing the area of the trapezoid. 2x - 3 4 Answer 3x + 4 A = 1/2 * (2x - 3 + 3x + 4) * 4 = 2 * (5x + 1) = 10x + 2 in2

You have enough paint to cover a surface of 20 ft2 You have enough paint to cover a surface of 20 ft2. The rectangular box (rectangular prism) you want to paint is 6 inches tall, 36 inches wide, and 24 inches deep. Do you have enough paint to cover the entire box? If yes, how much more area can you cover? Top Area Side Area Front Area Bottom Area Back Area

We will have enough to cover the box and an extra 3 ft2. You have enough paint to cover a surface of 20 ft2. The rectangular box (rectangular prism) you want to paint is 6 inches tall, 36 inches wide, and 24 inches deep. Do you have enough paint to cover the entire box? If yes, how much more area can you cover? Top Area Side Area Front Area Bottom Area Back Area 0.5 ft 2 ft 3 ft height = 6 in = 0.5 feet width = 36 in = 3 feet depth = 24 in = 2 feet Front = Back (3)(0.5) = 1.5 2(1.5) = 3 Top = Bottom (3)(2) = 6 2(6) = 12 Side = Side (2)(0.5) = 1 2(1) = 2 17 ft2 We will have enough to cover the box and an extra 3 ft2.

Volume V = Bh V = Bh V = πr2h Prism Cylinder Pyramid Cone Sphere Base height L w h base height r V = Bh V = πr2h V = Bh Square Base (B) Slant Height (l ) Pyramid's Height (h) r height Slant Height l V = 1/3 Bh V = 1/3 π r2 h V = 1/3 Bh r V = 4/3 π r3

Does a prism need to be a right prism for the volume formula to work? Think of a ream of paper Stacked nicely it has 500 sheets. If the stack is fanned, it still has 500 sheets.

Example: Find the volume of the box with length 2, width 6, and height 5.

Example: The volume of a box is 48 ft3 Example: The volume of a box is 48 ft3. If the height is 4ft and width is 6ft, what is the length.

Example: Find the volume of the cylinder with radius 4 and height 11.

Example: Find the volume of the pyramid. 6 4 5

Example: Find the volume of the pyramid. 5 8 8

Example: Find the volume of the cone. r= 7 9

Example: Find the volume of the cone. 12 r= 4

Example: Find the volume of the sphere. 9

Example: Find the volume of the sphere. Great Circle: A=25π u2

17 What is the volume of a box with edges of 4, 5, and 7?

18 If the volume of a box is 64 u3 and has height 8 and width 4, what is the length?

19 Find the volume of the cylinder with radius 6 and height 8.

The height of a cylinder doubles, what happens to the volume? 20 The height of a cylinder doubles, what happens to the volume? A Doubles B Quadruples C Depends on the cylinder D Cannot be determined

Find the volume of the pyramid. 21 Find the volume of the pyramid. 8 6 6

What is the volume of the cone? 22 What is the volume of the cone? 8 d=10

Find the volume of the sphere. 23 Find the volume of the sphere. 4

Find the volume of the sphere. 24 Find the volume of the sphere. Great Circle: A= 16π u2