SEGMENT ADDITION This stuff is AWESOME!

Can you see a shark? What about now?

NOTATION AB means the line segment with endpoints A and B. AB means the distance between A and B. A B AB = 14 cm

BETWEEN D G E C E is between C and D. G is not between C and D. For one point to be between two other points, the three points must be collinear. C

SEGMENT ADDITION POSTULATE If Q is between P and R, then PQ + QR = PR. What does this mean? Start with a picture: P R Q If point Q is between points P and R, then the distance between P and Q plus the distance between Q and R is equal to the distance between P and R.

SEGMENT ADDITION POSTULATE If PQ + QR = PR, then Q is between P and R. What does this mean? If the measure of segment PQ plus the measure of segment QR is equal to the measure of segment PR, then point Q must be between points P and R. 12 3 PR = 15 P R Q 14 3 Q P 15 R

COLORED NOTE CARD Segment Addition Postulate #2 If Q is between P and R, then PQ + QR = PR. If PQ + QR = PR, then Q is between P and R. P R Q

Q is between R and T. RT = 18 and QR = 10. Find QT. N is between L and P. LN = 14 and PN = 12. Find LP. L 14 N 12 P Q is between R and T. RT = 18 and QR = 10. Find QT. 10 R Q T 18

Find MN if N is between M and P, MN = 3x + 2, NP = 18, and MP = 5x. M 3x + 2 N 18 P 5x MN = 3 (10 ) + 2 MN = 32 3x + 2 + 18 = 5x 3x + 20 = 5x -3x -3x 20 = 2x 2 2 10 = x

CLASSWORK / HOMEWORK

Find and label the coordinates of the vertex C. THE DISTANCE AND MIDPOINT FORMULAS Investigating Distance: 1 Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse. B (6, 4) A (2, 1) x y 2 Find and label the coordinates of the vertex C. 3 Find the lengths of the legs of ABC. AB 5 yB – yA 3 4 – 1 4 Use the Pythagorean theorem to find AB. Remember: a 2 + b 2 = c 2 xB – xA 6 – 2 4 C (6, 1) 4 2 + 3 2 = c 2 16 + 9 = c AB = 5 16 + 9 = c 2 25 = c

Finding the Distance Between Two Points The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2). Using the Pythagorean theorem x 2 – x 1 y2 – y1 d x y C (x 2, y 1 ) B (x 2, y 2 ) A (x 1, y 1 ) a 2 + b 2 = c 2 You can write the equation (x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2 Solving this for d produces the distance formula. THE DISTANCE FORMULA The distance d between the points (x 1, y 1) and (x 2, y 2) is d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

Finding the Distance Between Two Points Find the distance between (1, 4) and (–2, 3). SOLUTION To find the distance, use the distance formula. d = (x 2 – x 1) 2 + ( y 2 – y 1) 2 Write the distance formula. = (x 2 – x 1) 2 + ( y 2 – y 1) 2 Substitute. –2 – 1 3 – 4 = 10 Simplify.  3.16 Use a calculator.

Applying the Distance Formula A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a goal line. The ball lands 45 yards from the same goal line and 40 yards from the same sideline. How far was the ball kicked? SOLUTION The ball is kicked from the point (10, 5), and lands at the point (40, 45). Use the distance formula. d = (40 – 10) 2 + (45 – 5) 2 = 900 + 1600 = 2500 = 50 The ball was kicked 50 yards.

( ) Finding the Midpoint Between Two Points x 1 + x 2 2 y 1 + y 2 , The midpoint of a line segment is the point on the segment that is equidistant from its end-points. The midpoint between two points is the midpoint of the line segment connecting them. THE MIDPOINT FORMULA x 1 + x 2 2 ( ) y 1 + y 2 , The midpoint between the points (x 1, y 1) and (x 2, y 2) is

( ) ( ) ( ) ( ) ( ) Finding the Midpoint Between Two Points x 1 + x 2 Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result. x 1 + x 2 2 ( ) y 1 + y 2 , Remember, the midpoint formula is . SOLUTION –2 + 4 2 ( ) 3 + 2 , 2 ( ) 5 , = 1 ( ) 5 2 , = The midpoint is , . 1 ( ) 5 2

( ) Finding the Midpoint Between Two Points CHECK Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result. CHECK From the graph, you can see that the point , appears halfway between (–2, 3) and (4, 2). You can also use the distance formula to check that the distances from the midpoint to each given point are equal. ( ) 1 5 2 (–2, 3) (1, ) 5 2 (4, 2)

( ) ( ) Applying the Midpoint Formula SOLUTION 1 You are using computer software to design a video game. You want to place a buried treasure chest halfway between the center of the base of a palm tree and the corner of a large boulder. Find where you should place the treasure chest. SOLUTION (25, 175) 1 Assign coordinates to the locations of the two landmarks. The center of the palm tree is at (200, 75). The corner of the boulder is at (25, 175). (112.5, 125) (200, 75) 2 Use the midpoint formula to find the point that is halfway between the two landmarks. 25 + 200 2 ( ) 175 + 75 , 225 2 ( ) 250 , = = (112.5, 125)

ANGLES

You will learn to classify angles as acute, obtuse, right, or straight.

What is an angle? Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.

Here are some examples of angles.

We can identify an angle by using a point on each ray and the vertex We can identify an angle by using a point on each ray and the vertex. The angle below may be identified as angle ABC or as angle CBA; you may also see this written as <ABC or as <CBA. The vertex point is always in the middle.

Angle Measurements We measure the size of an angle using degrees. Here are some examples of angles and their degree measurements.

Acute Angles An acute angle is an angle measuring between 0 and 90 degrees. The following angles are all acute angles.

Obtuse Angles An obtuse angle is an angle measuring between 90 and 180 degrees. The following angles are all obtuse.

Right Angles A right angle is an angle measuring 90 degrees. The following angles are both right angles.

Straight Angle A straight angle is 180 degrees.

Adjacent, Vertical, Linear Pair Supplementary, and Complementary Angles

Adjacent angles are “side by side” and share a common ray. 15º 45º

These are examples of adjacent angles. 45º 80º 35º 55º 130º 50º 85º 20º

These angles are NOT adjacent. 100º 50º 35º 35º 55º 45º

When 2 lines intersect, they make vertical angles. 75º 105º 105º 75º

Vertical angles are opposite one another. 75º 105º 105º 75º

Vertical angles are opposite one another. 75º 105º 105º 75º

Vertical angles are congruent (equal). 150º 30º 150º 30º

Linear Pair are adjacent angles that add to be 180 degrees. 75º 105º 105º 75º

Supplementary angles add up to 180º. 40º 120º 60º 140º Linear Pair: Adjacent and Supplementary Angles Supplementary Angles but not Adjacent

Complementary angles add up to 90º. 30º 40º 50º 60º Adjacent and Complementary Angles Complementary Angles but not Adjacent

1.6 Classify Polygons

Identifying Polygons Formed by three or more line segments called sides. It is not open. The sides do not cross. No curves. NOT POLYGONS POLYGONS

Terms Convex: a polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. Concave: a polygon that is nonconvex.

Classifying Polygons 3 8 4 9 5 10 6 12 7 n Number of Sides Type of Polygon Number of Side 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n n-gon

Definitions n-gon: a polygon with n number of sides. Equilateral: a polygon whose sides are all congruent. Equiangular: a polygon whose angles are all congruent. Regular: a polygon whose sides are equilateral and whose angles are equiangular.

Determine if the figure is a polygon. If yes, state whether it is convex or concave Yes, conclave Yes, convex

This figure is equilateral because all sides are the congruent It is also equiangular because all angles are congruent. Therefore this is a regular pentagon.

1.7 Find Perimeter, Circumference, and Area

Formulas Square Rectangle Triangle Circle w l s r 2 a c h d b Area: s2 Area: lw w Perimeter: 4s Perimeter: 2l + 2w l s Area: πr2 Area: bh r 2 a c h d Circumference: 2πr b Perimeter: a + b + c

Find perimeter and area of the figure below. A = l x w 9 ft P = 2l + 2w A = 108 ft2 P = 42 ft 12 ft

Find the approximate area and circumference of the figure below C = 2πr or dπ 18 in A = 254.3 in2 C = 56.5 in

A triangle ABC has vertices A(2,5), B(4,1), and C(8,3) A triangle ABC has vertices A(2,5), B(4,1), and C(8,3). What is the approximate perimeter of ΔABC? Hint use the distance formula for AB, AC, and BC (d = √[(x2-x1)2 +(y2-y1)2]) AB = 4.47 AC = 6.32 BC = 4.47 P = 15.26