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Proofs with Perpendicular Lines 3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units. Section 3.4 Proofs with Perpendicular Lines 147 hhs_geo_pe_0304.indds_geo_pe_0304.indd 147147 11/19/15/19/15 99:25:25 AAMM 3.4 Lesson WWhathat YYouou WWillill LLearnearn Find the distance from a point to a line. Construct perpendicular lines. Core VocabularyVocabulary Prove theorems about perpendicular lines. distance from a point to a line, Solve real-life problems involving perpendicular lines. p. 148 perpendicular bisector, p. 149 Finding the Distance from a Point to a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line. For example, the distance between point A and line k is AB. A k B distance from a point to a line Finding the Distance from a Point to a Line Find the distance from point A to ⃖ BD៮៮⃗ . y A(−3, 3) 4 D(2, 0) −4 4 x REMEMBER C(1, −1) B(−1, −3) Recall that if A(x1, y1) −4 and C(x2, y2) are points in a coordinate plane, then the distance SOLUTION between A and C is ——— — 2 2 Because AC ⊥ ⃖ BD៮៮⃗ , the distance from point A to ⃖ BD៮៮⃗ is AC. Use the Distance Formula. AC = √ (x2 − x1) + (y2 − y1) . ——— — — 2 2 2 2 AC = √ (−3 − 1) + [3 − (−1)] = √ (−4) + 4 = √ 32 ≈ 5.7 So, the distance from point A to ⃖ BD៮៮⃗ is about 5.7 units. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 1. Find the distance from point E to ⃖ FH៮៮⃗ . y 4 F(0, 3) G(1, 2) H(2, 1) −4 2 x −2 E(−4, −3) 148 Chapter 3 Parallel and Perpendicular Lines hhs_geo_pe_0304.indds_geo_pe_0304.indd 148148 11/19/15/19/15 99:25:25 AAMM Constructing Perpendicular Lines Constructing a Perpendicular Line P Use a compass and straightedge to construct a line perpendicular to line m through point P, which is not on line m. m SOLUTION Step 1 Step 2 Step 3 P P P m m A B AB m A B Q Q Draw arc with center P Place the Draw intersecting arcs Draw an Draw perpendicular line Draw compass at point P and draw an arc arc with center A. Using the same ⃖PQ៮៮⃗ . This line is perpendicular to that intersects the line twice. Label radius, draw an arc with center B. line m. the intersections A and B. Label the intersection of the arcs Q. — n The perpendicular bisector of a line segment PQ is the line n with the following two properties. — • n ⊥ PQ P M Q • n passes through the midpoint M of PQ — . Constructing a Perpendicular Bisector Use a compass and straightedge to construct A B the perpendicular bisector of AB — . SOLUTION Step 1 Step 2 Step 3 in. 1 2 3 4 5 6 A B A B A M B cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Draw an arc Place the compass Draw a second arc Keep the same Bisect segment Draw a line at A. Use a compass setting that is compass setting. Place the compass through the two points of greater than half the length of AB — . at B. Draw an arc. It should intersect intersection. This line is the Draw an arc. the other arc at two points. perpendicular bisector of AB— . It passes through M, the midpoint — of AB . So, AM = MB. Section 3.4 Proofs with Perpendicular Lines 149 hhs_geo_pe_0304.indds_geo_pe_0304.indd 149149 11/19/15/19/15 99:25:25 AAMM Proving Theorems about Perpendicular Lines TTheoremsheorems Theorem 3.10 Linear Pair Perpendicular Theorem If two lines intersect to form a linear pair of g congruent angles, then the lines are perpendicular. If ∠l ≅ ∠2, then g ⊥ h. 12 h Proof Ex. 13, p. 153 Theorem 3.11 Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one j of two parallel lines, then it is perpendicular to the h other line. If h ʈ k and j ⊥ h, then j ⊥ k. k Proof Example 2, p. 150; Question 2, p. 150 Theorem 3.12 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the m n same line, then they are parallel to each other. p If m ⊥ p and n ⊥ p, then m ʈ n. Proof Ex. 14, p. 153; Ex. 47, p. 162 Proving the Perpendicular Transversal Theorem Use the diagram to prove the j Perpendicular Transversal Theorem. 1 2 h SOLUTION 3 4 Given h ʈ k, j ⊥ h 5 6 k 7 8 Prove j ⊥ k STATEMENTS REASONS 1. h ʈ k, j ⊥ h 1. Given 2. m∠2 = 90° 2. Def nition of perpendicular lines 3. ∠2 ≅ ∠6 3. Corresponding Angles Theorem (Theorem 3.1) 4. m∠2 = m∠6 4. Def nition of congruent angles 5. m∠6 = 90° 5. Transitive Property of Equality 6. j ⊥ k 6. Def nition of perpendicular lines MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com 2. Prove the Perpendicular Transversal Theorem using the diagram in Example 2 and the Alternate Exterior Angles Theorem (Theorem 3.3). 150 Chapter 3 Parallel and Perpendicular Lines hhs_geo_pe_0304.indds_geo_pe_0304.indd 150150 11/19/15/19/15 99:25:25 AAMM Solving Real-Life Problems Proving Lines Are Parallel The photo shows the layout of a neighborhood. Determine which lines, if any, must be parallel in the diagram. Explain your reasoning. sstut u p q SOLUTION Lines p and q are both perpendicular to s, so by the Lines Perpendicular to a Transversal Theorem, p ʈ q. Also, lines s and t are both perpendicular to q, so by the Lines Perpendicular to a Transversal Theorem, s ʈ t. So, from the diagram you can conclude p ʈ q and s ʈ t. MMonitoringonitoring PProgressrogress Help in English and Spanish at BigIdeasMath.com Use the lines marked in the photo. ab c d 3. Is b ʈ a? Explain your reasoning. 4. Is b ⊥ c? Explain your reasoning. Section 3.4 Proofs with Perpendicular Lines 151 hhs_geo_pe_0304.indds_geo_pe_0304.indd 151151 11/19/15/19/15 99:25:25 AAMM 3.4 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Conceptp Check 1. COMPLETE THE SENTENCE The perpendicular bisector of a segment is the line that passes through the ________ of the segment at a ________ angle. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Find the distance from point X to line ⃖ WZ៮៮⃗. X(−3, 3) y Z(4, 4) 4 Find XZ. Y(3, 1) −4 −2 2 4 x Find the length of XY — . −2 W(2, −2) Find the distance from lineℓto point X. −4 MMonitoringonitoring ProgressProgress andand ModelingModeling withwith MathematicsMathematics In Exercises 3 and 4, f nd the distance from point A CONSTRUCTION In Exercises 5–8, trace line m and point to ⃖ XZ៮៮⃗ . (See Example 1.) P. Then use a compass and straightedge to construct a line perpendicular to line m through point P. 3. y Z(2, 7) 6 5. P 6. P 4 m Y(0, 1) m A(3, 0) −2 4 x X(−1, −2) 7. 8. P P m 4. y A(3, 3) m 3 1 x In Exercises 9 and 10, trace AB— . Then −3 1 CONSTRUCTION −1 use a compass and straightedge to construct the Z(4, −1) — −3 perpendicular bisector of AB . Y(2, −1.5) X(−4, −3) 9. 10. A B B A 152 Chapter 3 Parallel and Perpendicular Lines hhs_geo_pe_0304.indds_geo_pe_0304.indd 152152 11/19/15/19/15 99:25:25 AAMM ERROR ANALYSIS In Exercises 11 and 12, describe and In Exercises 17–22, determine which lines, if any, must correct the error in the statement about the diagram. be parallel. Explain your reasoning. (See Example 3.) 11. 17. v w 18.
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