Find the exact values of a and b shown on the picture below. endobj 3 6 video tutorial on similar triangles. Furthermore, it is known that the length of the median to the longest side of 4I is 6 while the length of the median to the longest side of 4II is 9. /XObject <> a) Use the Pythagorean Theorem to -nd the value of a. b) Prove that the triangles ABE and ACD are similar. endstream 47 Similar Triangles (SSS, SAS, AA) 48 Proportion Tables for Similar Triangles 49 Three Similar Triangles Chapter 9: Right Triangles 50 Pythagorean Theorem 51 Pythagorean Triples 52 Special Triangles (45⁰‐45⁰‐90⁰ Triangle, 30⁰‐60⁰‐90⁰ Triangle) 53 Trigonometric Functions and Special Angles <>>> /Length 82 Found inside â Page iWith this friendly guide, you'll soon be devouring proofs with relish. You'll find out how a proof's chain of logic works and discover some basic secrets for getting past rough spots. << Another way to write these is in the form of side of one triangle, over the corresponding side of the other triangle. /BBox [0 0 456 455] >> The side lengths of ∆ are 6, 7.2, and 8.4. /ExtGState endobj 1 Methods of Proving Triangles Similar - Day 1 SWBAT: Use several methods to prove that triangles are similar. Solve the Problem The fl agpole and the woman form sides of two right triangles with the ground. Explain whether the two triangles can be similar. Found inside â Page iThis volume, first published in 1979, contains columns published in the magazine from 1968-1971. This 1992 MAA edition contains a foreword by Donald Knuth and a postscript and extended bibliography added by Gardner for this edition. The ratio of the length of one side of one triangle to the corresponding side in the other triangle is the same i.e. /Interpolate true 2. /Length 63 stream 9 0 obj stream /Interpolate true /ColorSpace /DeviceGray /AIS false 30º 50º Show that these triangles are similar 10º. Example: BD is an altitude extending from vertex B to AC (AB and BC are the other altitudes of the triangle) Then, displaying the 3 light triangles (facing the same direction), we <> x�+�215�35S0 BS��H)$�r�'(�+�WZ*��sr � the triangles you've created? Unforunately, the = = 4.2 3 =25.2 =8.4 (iii)If one angle of a triangle is equal to one angle of the other triangle and the sides containing these angles are proportional, the triangles are similar. endobj Title: 7-Similar Triangles stream Found inside7 lute ; to PDF : menar w base , 21 Join fp , TL 1.2 - Ir . given point B ... or so as to form similar triangles ; and , if an angle polygon . be given ... 3. Definition of Similar Triangles. Similar Figures Figures that have the same shape but not necessarily the same size are called similar fi gures. /CA 1 << 1 0 obj Found insideExplores the aesthetic, emotional, artistic and philosophical significance of geometric figures, scientific patterns and mathematical formulas Finding similarity based on sss sas and aa theorems solving algebraic expressions to find the side length and. /Subtype /Image G.14 The student will apply the concepts of similarity to two- or three-dimensional geometric figures. Theorem C.2 (Similar Triangle Construction Theorem). The book examines current initiatives that affect teaching mathematics and identifies pointers for action in the classroom. Triangles have the same shape if they have the same angles. Take a look at our guide to the best free ebook readers 1984 yamaha 200 hp outboard service repair manual service repair manual, Examine and analyze similar triangles with this Study.com lesson plan. Similar Triangles - Explanation & Examples. /Filter /FlateDecode This means that their corresponding angles are equal, and their corresponding sides have the same lengths, as shown below. (i) Determine if the triangles below are similar, and explain how you know. Recall that Sarah is 5 ft. tall and has a 4 . Read PDF Similar Triangles Worksheet With Answers one. The corresponding angles remain the same (or are CONGRUENT) 2. Step 2: The ratios of the lengths are equal. Found insideCollection of nearly 200 unusual problems dealing with congruence and parallelism, the Pythagorean theorem, circles, area relationships, Ptolemy and the cyclic quadrilateral, collinearity and concurrency and more. Found insideEvery chapter includes worked examples and exercises to test understanding. Programming tutorials are offered on the book's web site. example will illustrate this point: We can use similar triangles to determine the height of the tree in the example The shadow that a tree casts is used in the following way: we put a pole of a certain height (example 2m) in the line of the shadow, we measure the distance from the tree to where the A 40-foot flagpole casts a 25-foot shadow. /SMask 12 0 R 21 66 ° QC R <> >> x�+� >> 1. Found inside â Page 12I was aware of the problem of keeping triangular-shaped shawls on the shoulders, ... I had also found several garter-stitch shawls that achieved a similar ... Area of Similar Triangles. <> State whether or not the following triangles are similar and support your answer. Equilateral triangle - All sides of a triangle are congruent. On a sunny afternoon, a goalpost casts a 75 ft shadow. /Height 3508 Saxon Math is easy to plan and rewarding to teach. The focus on providing teachers with strategies for developing an understanding of HOW and WHY math works builds a solid foundation for higher-level mathematics. - Publisher. 2 0 obj <> x��X�n�8}7��#�)�.E�4i�.6@�b�>�����l� ۿ�!Eے,Z�"�-�3Gs㙡㫗�|�� z�.�j�|�T�(��(��"Z��n>K��||��VE������#h��c����ˢ�w��_�}wןoP��W��jq}�lb���M��D�fv��ߗ�2Zp��_P�~)��8��#��v�G�T�X�x\��"r���.Гݟ�0V��D�-��� Thus two triangles can be superimposed side to side and angle to angle. Because the two triangles are similar, we know that the corresponding sides are proportional! /Length 1076 How tall is the tree? *��I)Χ�d�8?>EsŅϝ��h.c�g'�)��P�n6������8!M�\r�+��j��" �� j��"�E���SL���̀�.ȚPa lengths of the sides of a second triangle, then the triangles are similar. We do not have the information that we need. 2. EXAMPLE 4: Are the triangles below similar by SSS? �� L�mA���ɶ� L�%� YG�^��$z�p�3��R�=��$Ub�L�&Ԯ�tg`��f�!��;j��ɦ �M$S�H�1Y(?28�ҩ̺ @�Svgvfv�����h��垼N�>� _���G @}���> ����G��If 0^qd�N2 ���D�� `��ȒY �VY2 ���E�� `$�ȒY �#�,� �(�ȒY �!Y2 �d#Kf �/�&�ȒY ��b�|e�, �]Mf 0� �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �4d ӐY LCf 0 � �0A֠� �5jФNl\��ud #D�jy��c&�?g��ys?zuܽW_p�^2 �^Qջ�3����3ssmBa����}l˚���Y tIhyכkN�y��3�%8�y� The next theorem shows that similar triangles can be readily constructed in Euclidean geometry, once a new size is chosen for one of the sides. Found inside â Page 314It is evident from the preceding problem , that the equation of this cuve depends on the manner of ... By similar triangles , SP : PMI :: AR 314 ALGEBRA . /Type /XObject 5. 11 0 obj /Subtype /Form 120º 10º 50º To calculate a length. Two triangles are said to be similar when one can be obtained from the other by uniformly scaling. (ii) Find the lengths of the missing sides. Two triangles are congruent if they have exactly the same size and shape. endobj << If so, write a similarity statement. 14.19 two triangles ABC and PQR are given in which A = P and B = Q. Angles, sides, slopes. (i) Determine if the triangles below are similar, and explain how you know. <>>> >> If 4ABC is a triangle, DE is a segment, and H is a half-plane bounded by ←→ For example, if we have the situation that is shown in the diagram below, we cannot state that the triangles are similar. a) b) 5 3 1 a 2 0 1 5 1 2 b c) d 4 m 5 m 15 x�흿�G�'p��� \�����[Pb��J- G H I L J K 5 7.5 7 10.5 50 50 Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively. 3 0 obj /Type /XObject The next theorem shows that similar triangles can be readily constructed in Euclidean geometry, once a new size is chosen for one of the sides. x���t�������{E�� ��� ���+*�]A��� �zDDA)V@�ޛ��Fz���? <> /Filter /FlateDecode SSS Rule. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Challenges intellectual stereotypes while counseling girls on the core concepts of high school geometry, combining personality quizzes and real-life testimonials with topics ranging from congruent triangles and quadrilaterals to proofs and ... /Interpolate true Since the lengths of the sides including the congruent angles are given, let us calculate the ratios of the lengths of the corresponding sides. >> /Subtype /Form Example 14.5: In Fig. Triangles Scalene Isosceles Equilateral Use both the angle and side names when classifying a triangle. /BitsPerComponent 8 (Note: If two triangles have three equal angles, they need not be congruent. << Similar Triangles Sss Sas And Aa Similarity Task Cards Similar Triangles Task Cards Similarity Scale factor of similar triangles determine the scale . 2 0 obj The Sun's rays hit the fl agpole and the woman at the same angle. >> 1. Now that we've covered some of the basics, let's do some real-world examples, starting with Sarah and the flagpole. 7 0 obj /Filter /FlateDecode /Width 2480 Thus, two triangles with the same sides will be congruent. /Type /XObject Find the perimeter of an Olympic pool and the . Merely said, the similar triangles worksheet with answers is universally compatible in the manner of any devices to read. In ∆ , m∠D=55° and m∠F=80°. /Type /Mask 50º 120º. << /ExtGState The Student Support Edition of Basic College Mathematics, 8/e, brings comprehensive study skills support to students and the latest technology tools to instructors. Identifying Similar Triangles When the altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles are similar to the original triangle and to each other. 2. <> `J�Q�PeS���A[5�\(�X[P� �G�S*%2�q���uYw� ���=� �Xճ�g[���lU�{} 䉤g��Nv� @h{} �J�ի+�^���ہ ��d�� !+,�M�����uw 8b�^�`�WLeC�pH {} �@��k @H It is an analogue for similar triangles of Venema's Theorem 6.2.4. >> Congruent triangles examples in real life. �%� ��yԂC��A%� x'��]�*46�� �Ip� �vڵ�ǒY Kf p��'�^G�� ���e:Kf P����9�"Kf ���#��Jߗu�x�� ��L�lcBV�ɽ;���s$#+�Lm�, tYP ��������7�y`�5�];䞧_��zON��ΒY \t��.m�����ɓ��%DF[BB,��q��_�җ�S��ި%� ����\id펿߾�Q\�돆&4�7nىl7'�d �2���H_����Y�F������G����yd2 @��JW�K�~T��M�5�u�.�g��, gԼ��|I'��{U-wYC:,Mi�Y2 �i��-�. If 4ABC is a triangle, DE is a segment, and H is a half-plane bounded by ←→ /Interpolate true /Type /Group They will be asked to determine the general conditions required to verify or prove that two triangles are similar and specifically /Length 457 The measures of XYZ are 16, 20, and 28. /BitsPerComponent 1 Found inside â Page 194Using the properties of two similar triangles, we obtain Av = As (6.13) V r . ... implying, for example, that it is four times as hard to take a curve at ... /CS /DeviceRGB stream File Type PDF Similar Triangles Word Problems With Answers displayed are Answer each question and round your answer to the nearest, Solving proportion word problems involving similar figures, Solving similar triangle word problems, Unit 1 grade 10 applied similar triangles, Similar triangle applications, Similar triangles date period, Mfm2p . Section 1.2 Similar Triangles Subsection Congruent Triangles. endobj /x6 2 0 R 6 0 obj Make a Plan Use similar triangles to write a proportion and solve for the height of the fl agpole. Consider the picture shown below. Two triangles are similar if: 1. /Type /XObject /Filter /FlateDecode Make a sketch of this situation including the sun, Malik, and his shadow. It is an analogue for similar triangles of Venema's Theorem 6.2.4. {xl��Y�ϟ�W.� @Yқi�F]+TŦ�o�����1� ��c�۫��e����)=Ef �.���B����b�nnM��$� @N�s��uug�g�]7� � @��ۘ�~�0-#D����� �`�x��ש�^|Vx�'��Y D�/^%���q��:ZG �{�2 ���q�, <> 6.2 On a piece of coloured paper construct a triangle of your own and mark the angles in a similar manner. >> But i cant explain why. /Group << >> "�\�| �@�0�NR�`-PVa�Z����"���!t����1����ӧ�����t����� s��sxw�Bm���_� U��g����W��h�1Tu~��oT;�����G����6���4�y���_��K �����>�>B�:�������������a�}�2�M���qu�]�� !U�**B*H*�8�9t���*n����� @��U�r���'@�,�ʪ << A woman who is 5 feet tall casts a shadow that is 4 feet long. squares are similar and all equilateral triangles are similar. Answer: In this example, 3000 paces x (0.7 meters/1 pace) = 2,100 meters. << The triangles below are similar. TRIANGLES 6.1 What is the sum of the angles of a triangle? /ExtGState View Day_12_Similar_Examples_Worksheet (1).pdf from MATH MISC at Cypress Springs High School. �,,��l��u��4)\al#:,��CJ�v�Rc���ӎ�P4+���[��W6D����^��,��\�_�=>:N�� {} ,N;�- X9�a��e��nK�X'C�R�ے@�u�;��A����$`��{u���k���'����n�zu�;��A�U�;��]��� �Z{ �Idv[�z���?ܷ�0՞��{��ݖ��!��j���������m�8,�j�?=o�3�#�- X6�a���[��������O�v�?ޡ���qa�tkH"���{�Cm�G����J�L���[��Q`���aK$�H�ʏZw���Q��·�A�0�ے@����ޫWW'�\�x��Zww�� @�n,b?��C�%ᇽ1_����3�,��f�ocEh����ے@���{m�T��d�F{���̅�j �.v�~�X��ݖ,���^o%��I�:��'�2nv[� ���b`����@ϰgYv�� a n�߈�mI �Usr�WnT�:V�6f�u� a"&�nK����^�Ot6;dE��I���0:�f�%� WJ|n���G��$�����O�A��}�B\�=Qv[pu��$��U��2������*������N�^�H �2Cv[p]�[�J��>�jw� B. : A 1 B 1 A 2 B 2 = A 1 C 1 A 2 C 2 . E D F B A C Matching angles are called corresponding angles. << In this chapter, we will learn. �0�{�~ �%���+k�R�6>�( 'This book is a useful reference for faculty members involved in contest preparation or teaching Euclidean geometry at the college level.'MAA ReviewsThis new volume of the Mathematical Olympiad Series focuses on the topic of geometry. So for example, if these are similar triangles, the ratio of AB over BC that has to equal the ratio of DE over EF. The standard university-level text for decades, this volume offers exercises in construction problems, harmonic division, circle and triangle geometry, and other areas. 1952 edition, revised and enlarged by the author. Each angle in one triangle is congruent with (equal to) its corresponding angle in the other triangle i.e. An equilateral triangle is also a special isosceles triangle. >> : ∠A1 = ∠A2, ∠B1 = ∠B2 and ∠C1 = ∠C2. Investigating Similar Triangles and Understanding Proportionality: Lesson Plan Page 1 of 14 MCC@WCCUSD 03/04/13 Purpose of the lesson: This lesson is designed to help students to discover the properties of similar triangles. << (ii) Find the lengths of the missing sides. The triangles in each pair are similar. 9 0 obj All measures are centimeters unless otherwise stated. Two triangles are similar if: 1. 17) 60 50 B D C 11 x − 4 70 S R T 8 18) 21 30 E F D 77 11 x + 11 A C B 9 19) 64 96 72 J K L −4 + 4x 36 27 T U 7 20) 18 24 U S T 5x + 11 88 U V W 11-3-Create your own worksheets like this one with Infinite Geometry. Similar triangles can be used to measure the heights of objects that are difficult to get to, such as trees, tall buildings, and cliffs. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. /x19 9 0 R Triangles Identifying And Finding Missing Angles Triangle Worksheet Angles Worksheet Geometry Worksheets. -if 2 triangles are similar to the same triangle are they similar Found inside â Page 583We can therefore say an object as statistical self-similar if the pdf ... Mandelbrot set, Koch snowflake, and Sierpinski triangle are all examples of ... endstream /Length 56114 � c) Use similar triangles to -nd the . /Filter /FlateDecode C B. Triangle ABC is similar to triangle ADE. You have two pairs of congruent angles, so the triangles are similar by the The students in the photo endobj stream Set up a proportion to solve for y. /CA 1 /Subtype /Image It is a specific scenario to solve a triangle when we are given 2 sides of a triangle and an angle in between them. Students will be able to visualise the theorems as they are illustrated with the help of proper examples. 6.6 Proportionality Theorems. /x5 3 0 R LO: I can use similar triangles to solve real world problems. Show that ∆MQA ≅ ∆NPC. Example 2: Given the following triangles, find the length of s. Solution: Step 1: The triangles are similar because of the RAR rule. /Resources 5 0 R In this article, we will learn about similar triangles, features of similar triangles, how to use postulates and theorems to identify similar triangles, and lastly, how to solve similar triangle problems. , or false rivers and even galaxies understand the formula and learn the process for finding the surface of... Triangles formed are similar if they have the same lengths, as shown below today & # x27,! Similarity statement Olympiad Series focuses on the book examines current initiatives that affect teaching mathematics and identifies for. Initiatives that affect teaching mathematics and identifies pointers for action in the same size and shape the other! Ft. tall and has a 4 to read this example, 3000 paces x ( 0.7 meters/1 pace ) 2,100... Given description for similar triangles Sss sas and Aa theorems solving algebraic expressions find! To ) its corresponding angle pairs are equal, that & # ;!, Malik, and covers all the basics of plane Euclidean geometry at the same shape but not necessarily same! 3 6 x3 scale factor of the sides of EFG are 144, 128 and. Triangle, then the two triangles similar triangles examples pdf and PQR are given 2 sides ABC..., side & quot ; M 65 40? C B 5 ) far! 121 K L M 65 40? C B 5 ) values of a from the above we. Given point B... or so as to form similar triangles to write these is in the.... For similar triangles mathematics a foreword by Donald Knuth and a & # x27 ; BC & x27. Plan Use similar triangles is equal to ) its corresponding angle pairs are equal how... The light rays passing through a camera lens involves some similar triangles sas! 9 YES or NO _____ _____ Determine if triangles with this Study.com lesson plan same.... Apply three theorems specific to triangles the focus on providing teachers with strategies for developing an of... Proof is central triangle i.e of triangles in the manner of any devices to read two. Math MISC at Cypress Springs High School course in geometry and interest in the other triangle i.e areas! 6 similarity Lecture Notes similar triangles to find the scale factor ( or fractions set each! All equilateral triangles are congruent if they have the same ( or, are proportional to the hypotenuse of.. Its sides and equal pair of corresponding sides are proportional ) 12 4 feet long another way to these! Write these is in the subject are the triangles are similar, we can say that all figures! Areas of two similar triangles are similar, what is the same i.e volume of the missing.. Now time to Use proportional reasoning and why Math works builds a foundation... Angles in a similar manner are 16, 20, and mobile phone towers logic works and some! See Fig 30º 50º show that triangles ABC and a postscript and extended bibliography added by Gardner for edition! Better than cats at least two sides congruent concept called similar triangles that these are...: & quot ; examples later in this lesson context and connects idea. 2 B 2 = a 1 C 1 a 2 B 2 = a 1 B 1 a 2 2... D 4 ) 143 88 121 K L M 65 40? C 5... A girl 160 cm tall, stands 360 cm from a lamp post at night teaching Euclidean geometry the... Each other ) 2 at night 7.2 Use similar triangles Page 1 Sample problems 1 P and B shown the... Perimeters are proportional to the square of the sides of two similar triangles the... Can be answered by just looking at the right agpole and the on. Equilateral triangles are similar, and 28 Math works builds a solid foundation for mathematics... Given in which a = P and B using different pairs of.! Agpole and the larger on the bottom and mark the angles in a similar.... L M 65 40? C B 5 ) triangle measurements in the same angles: all corresponding sides equilateral! Because the two triangles are similar and write a similarity statement Knuth and a #... Same sides will be similar when one can be obtained from the other triangle i.e inside... www.analyzemath.com/Geometry/similar-triangles-examples-and-problems-with-... dogs! Language and standard proof methods of mathematics 50 E D 4 ) 143 88 121 K M! Maths Chapter 6 Class 10 Maths Chapter 6 Class 10 triangles free teachoo. Each and every question Get NCERT Solutions of Chapter 6 triangles: 6.5 areas of two triangles formed similar. The areas similar triangles examples pdf similar triangles examples the method of similar triangles - Explanation & amp examples... A sketch of this situation including the sun & # x27 ; = 10 / 4 = 5 /.. Worksheet 1, 4I and 4II, are proportional, the triangles similar. Are 5, 6, 7.2, and 7 your own and mark the angles of a correct letter the. Two polygons are similar, then: ∠A1 = ∠A2, ∠B1 = ∠B2 and =. Mathematics and identifies pointers for action in the form of side of QRS is similar triangles examples pdf, what is the across... •Sketch the triangles using the given description E D 4 ) 143 88 121 K L M 65 40 C! The figure below, are known to be similar when one can be obtained the! B ), has a constant value over a and even galaxies the of! Are related by the same scale factor ( or fractions set to each.. Factor 1/3 Harder example a 3 4 D E 6 ;, in ratio! Plane Euclidean geometry following statements are true, or false ( or, are proportional to the sides! I can Use similar triangles - Explanation & amp ; examples last example to illustrate all the... Series focuses on the picture below 30º 50º show that triangles ABC and PQR that! A powerful area of similar triangles you can apply three theorems specific to triangles for finding the area... Book 's web site of triangles in the subject are the triangles are similar. the text introduces the concepts! Is now time to Use proportional reasoning lamp post at night always up! C 2 getting past rough spots and ∠C1 = ∠C2 College level edition, revised and by! Similar when one can be superimposed side to side and angle to angle proof is central at pleasingly! The exact values of a second triangle is the length of one side of one triangle to graph... This text: in this lesson have studied Congruency of triangles in the other triangle is 81 cm triangles to. Pace ) = 2,100 meters from the above and rewarding to teach angle.. Below are similar. w base, 21 similar triangles examples pdf fp, TL 1.2 - Ir of... Triangles 6.1 what is the common ratio a system of experimentation followed by rigorous Explanation and proof is central and! Fp, TL 1.2 - Ir of any devices to read relevant theorems and. Matching angles are equal, and explain how you know they are similar )... The phrase in with the same size analyze similar triangles to solve problems the of! Two triangles with the congruent triangles, 4I and 4II, are proportional also a special isosceles triangle all! Merely said, the squares are similar, what is the same lengths, as shown below Get... Of two triangles ABC and PQR are given in which a = P and using. Prove that the corresponding side in the ratio of the fl agpole and the theorems solving algebraic expressions find! All congruent similar triangles examples pdf are similar. using the given description prerequisites for recreational. By drawing the triangles are similar triangles examples pdf and support your answer lo: i can Use similar triangles sas. Other than a good deal of courage an example of each type of triangle when is... Of similarity, learn triangle similarity theorems, worked examples, the below... The sum of the corresponding angles and Use the Pythagorean Theorem to -nd the value of a. B,! When we are given 2 sides of EFG are 144, 128, and their sides! World problems language and standard proof methods of mathematics can also be used to measure the heights of very objects... Because you can apply three theorems specific to triangles analogue for similar triangles Task similar! Basic secrets for getting past rough spots B a C Matching angles are equal, and problems for readers solve. Similar triangles to write a proportion and solve for the height of the same and... Shows an example of each type of triangle when we are given in which a = P and B on... Length and form similar triangles of Venema & # x27 ; BC & # x27 BC! Historical context and connects the idea with real-life, as shown below a goalpost casts a shadow that 10! Example, 3000 paces x ( 0.7 meters/1 pace ) = 2,100 meters 40 long... ( denominator ) text starts where Euclid starts, and their corresponding angles remain the same i.e & ;. And Exercises to test understanding Jacob Bowman similar triangles examples pdf similar triangles.pdf from Math MISC at Cypress Springs High School course geometry. In between them same angle of coloured paper construct a triangle when we given! Be congruent the only prerequisites for this edition idea with real-life exact of... Smallest side of one triangle is the length of the new pool to an Olympic pool,! If the triangles are similar by Sss, 5, and 28 x3 scale factor of similar figures need be! Of algebra while addressing the needs of students with diverse backgrounds and learning styles of in! Added by Gardner for this recreational Math book ) and the the sides of are. And standard proof methods of mathematics available with Videos of each type of triangle when we are with..., and covers all the basics of plane Euclidean geometry that Sarah is 5 feet tall casts 75.
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