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Applying Distance and Midpoint Formulas to Determine Segment Length

Learn how to apply and interpret the Distance and Midpoint formulas to find the length of line segments. This lesson also covers undefined terms in geometry and the Pythagorean Theorem.

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Applying Distance and Midpoint Formulas to Determine Segment Length

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  1. Learning Objective We will apply1 the Distance and the Midpoint formulas to determine the Segment2 Length. What are we going to do? What does apply & segment mean? Apply & segment means ___________. CFU Activate Prior Knowledge To find the length of a line segment, use the Pythagorean theorem to calculate the length of the hypotenuse. On your whiteboard, write the Pythagorean Theorem: Students, you already know the Pythagorean theorem. Now, we will apply the Distance and the Midpoint formulas to determine the Segment Length. Make Connection Find the length of the hypotenuse. 1 put to use 2 part of a line cut into small parts or sections. Vocabulary

  2. Concept Development The most basic figures in geometry are undefined1 terms, which cannot be defined using other figures. CFU • On your whiteboard, Identify the following: • . • . • . • . • E. • In your own words, describe the undefined terms. • In geometry, undefined terms are _____. Point Endpoint Line Ray plane 1 "three undefined terms of geometry" Vocabulary

  3. Concept Development Collinear means that points are on the same line, coplanar means “together on the same plane.” Note: co- usually means together or joint. For example, coauthors author a book together. Use the Distance Formula to determine the length of each segment in the figure shown.

  4. Concept Development You can use the Midpoint Formula to find the midpoint of a segment on the coordinate plane. On your whiteboard, determine the coordinates of the midpoint for each segment. Identify the quadrant that each midpoint lies in. 1. 2. II I

  5. Skill Development/Guided Practice 12 cm 9 cm 5 in 12 in The Pythagorean Theorem describes the relationship between the lengths of the sides of a right triangle. How did I/you know if the Pythagorean Theorem can be used? How did I/you identify the hypotenuse/legs of the triangle? How did I/you find the unknown side length? CFU Use the Pythagorean Theorem to find an unknown length of a side. 1 Determine2 if the Pythagorean Theorem can be used. Identify3 the hypotenuse and two legs of the right triangle. (record) Find the unknown length of the side using the Pythagorean Theorem. Substitute4 the two known values and solve. Interpret5 the solution. 1 2 2 leg2+leg2=hyp2 3 a 3 4 leg = leg = hyp = leg = leg = hyp = 12 cm 12 in 9 cm 5 in 15 cm ? cm ? in 13 in leg2+leg2=hyp2 leg2+leg2=hyp2 122 + 92 = hyp2 122 + 52 = hyp2 144 + 81 = hyp2 144 + 25 = hyp2 225 = hyp2 2 figure out 3 find (synonym) 4 replace (synonym) 5 explain Vocabulary 169 = hyp2 15 = hyp 13 = hyp √ √ √ √ hypotenuse hypotenuse 15 cm 13 in

  6. Skill Development/Guided Practice (continued) Solving Math Problems 1 Determine what the question is asking. 2 Determine the math concept required. 3 Determine relevant information. 4 Solve the problem, then interpret the answer. 5 Check the reasonableness of your answer. How did I/you determine what the question is asking? How did I/you determine the math concept required? How did I/you determine the relevant information? How did I/you solve and interpret the problem? How did I/you check the reasonableness of the answer? CFU 1 2 3 4 5 Solving Math Problems 1 What am I trying to find? 2 What do I know about this idea? 3 What amounts do I know? 4 Solve the problem or answer the question. 5 Did I answer all parts of the question? Roseville to Edgewood exits is 59 − 35 = 24 mi, so the distance from the Roseville to Freestone exits will be ½ (24) = 12 mi. Midtown to Roseville exits is 35 − 17 = 18 mi, so the distance from the Midtown to Freestone exits will be 18 + 12 = 30 mi.

  7. Relevance Soledad Salinas River 16mi King City 12mi The Pythagorean Theorem, Distance and the Midpoint formulas describes the relationship between the lengths of the sides of a right triangle. Using the Pythagorean Theorem will help you find the distance between locations. 1 How long is the Salinas River from King City to Soledad? hyp leg leg Using the Midpoint formula will help you do well on tests. 2 Does anyone else have another reason why it is relevant to use the Pythagorean Theorem, Distance and the Midpoint formulas? (Pair–Share) Why is it relevant to use the these formulas? You may give me one of my reasons or one of your own. Which reason is more relevant to you? Why? CFU

  8. A B 6 12 ? ? 6 10 d 6 cm 8 cm The Pythagorean Theorem describes the relationship between the lengths of the sides of a right triangle. Skill Closure Word Bank leg2+leg2=hyp2 Word Bank 62 + 82 = hyp2 Pythagorean Theorem right triangle side hypotenuse leg length leg = leg = hyp = 6 cm 10 cm = 36 + 64 = hyp2 8 cm 100 = hyp2 ? cm 10 = hyp hypotenuse 10 cm Access Common Core With which triangle could you use the Pythagorean Theorem to find the unknown length? Explain why you cannot use the Pythagorean Theorem on the other. I can use the Pythagorean Theorem on Triangle A because it is a right triangle. I cannot use the Pythagorean Theorem on Triangle B because it is not a right triangle. √ √ Summary Closure What did you learn today about applying the Pythagorean Theorem, Distance and the Midpoint formulas to determine the Segment Length? (Pair-Share) Use words from the word bank.

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