Grade 9 / Similar triangles 48 exercises (100% corrected)

a
MNP15o ABC65o ABC15m20m?aDEF10m?24mb ChingQuizz : 2 exercises available for Quizz assessment : 1. Around the triangles E.11233 Definition: in a triangle ABC : The side opposite an angle is the side that is not ad-jacent to that angle. If the triangle ABC is a right triangle at A , the hy-potenuse is the side opposite the right angle (the seg-ment [ BC ] . Proposition: In a triangle, the sum of the measures of its angles is always 180 o . Consider the triangle MNP , which is a right triangle at P : Complete the dotted lines : a The hypotenuse of the right triangle MNP is the seg-ment : : : b The angle measuring 15 o is named by the three letters . . . . c The side opposite angle MNP is segment : : : d Angle MNP measures : : : : : : o E.11236 Below is a right triangle ABC with C : 1 a Name the side opposite vertex B . b Name the vertex opposite side [ BC ] . c What does side [ AB ] represent in triangle ABC ? 2 Give the measure of angle ABC 3 a Draw the height of triangle ABC from vertex C . Note that H is the foot of this height. b Give the measure of angle ACH . E.11235 Consider a triangle whose three angles measure three consecutive integers. Give the measure of the largest angle. 2. About the Pythagorean theorem E.11237 Pythagorean theorem: if a triangle is right-angled, then the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Example: (finding the hypotenuse) Let MNP be a right triangle at M with MN =4 m and MP =3 m . According to the Pythagorean theorem, we have the follow-ing equality: NP 2 = MN 2 + MP 2 NP 2 = 4 2 + 3 2 NP 2 = 16 + 9 NP 2 = 25 NP = 25 = 5 For each of the triangles below, determine the length of the side whose measurement is unknown : Hints: 1 2 =1 ; 9 2 =81 ; 17 2 =289 ; 25 2 =625 ; 33 2 =1089 2 2 =4 ; 10 2 =100 ; 18 2 =324 ; 26 2 =676 ; 34 2 =1156 3 2 =9 ; 11 2 =121 ; 19 2 =361 ; 27 2 =729 ; 35 2 =1225 4 2 =16 ; 12 2 =144 ; 20 2 =400 ; 28 2 =784 ; 36 2 =1296 5 2 =25 ; 13 2 =169 ; 21 2 =441 ; 29 2 =841 ; 37 2 =1369 6 2 =36 ; 14 2 =196 ; 22 2 =484 ; 30 2 =900 ; 38 2 =1444 7 2 =49 ; 15 2 =225 ; 23 2 =529 ; 31 2 =961 ; 39 2 =1521 8 2 =64 ; 16 2 =256 ; 24 2 =576 ; 32 2 =1024 ; 40 2 =1600 https://chingmath.fr chapExoCorrec/11233 sacados/11233 MNP15o chapExoCorrec/11236 sacados/11236 ABC65o chapExoCorrec/11235 sacados/11235 chapExoCorrec/11237 sacados/11237 ABC15m20m?aDEF10m?24mb
IJK17m15m?aTUV25m?24mb ABC16m12m?D25m? BAC4m3mMN8m10m¸o˛o ABC100o50oDEF50o30o E.11238 Example: (Finding a side adjacent to the right angle) Let QRS be a right triangle at R with QS =13 m and RS =12 m . According to the Pythagorean theorem, we have the follow-ing equality: QS 2 = RQ 2 + RS 2 13 2 = RQ 2 + 12 2 169 = RQ 2 + 144 RQ 2 = 169 − 144 = 25 RQ = 25 = 5 m For each of the triangles below, determine the length of the segment whose measurement is unknown : E.11239 In the configuration below, trian-gles ABC and BCD are right triangles at A and C , respec-tively: Determine the length of segment [ CD ] . Hint: You must use the Pythagorean theorem twice. 3. About Thales’ theorem E.11234 Consider the two right triangles at B : ABC with AB =4 m ; AC =3 m AMN with AM =8 m ; MN = 10 m Furthermore, point B belongs to segment [ AM ] and point C belongs to segment [ AN ] . 1 a Using the Pythagorean theorem in triangle ANM , determine the length of segment [ AN ] . b Using the converse of Thales’ theorem, prove that ( BC ) == ( MN ) . Converse of Thales’ theorem: Let A , B , M be aligned and let A , C , N aligned such that AB AM = AC AN , then the lines ( BC ) == ( MN ) 2 a Using Pythagoras’ theorem in triangle ABC , deter-mine the length of segment [ BC ] . b What can you say about the lengths of triangle AMN compared to those of triangle ABC ? 3 Additional question: Justify that angles NMA and CBA are equal in measure. 4. Definition of similar triangles (AA case) E.8012 Definition: Two triangles are said to be similar if their angles are equal two by two. If two triangles are similar, two sides are said to be congruent if they are opposite angles of equal mea-sure. Consider the two triangles ABC and DEF : 1 Show that the two triangles ABC and DEF are similar. 2 Complete the sentences : a side [ AC ] is homologous to side . . . . . . b side [ DF ] is homologous to side . . . . . . https://chingmath.fr chapExoCorrec/11238 sacados/11238 IJK17m15m?aTUV25m?24mb chapExoCorrec/11239 sacados/11239 ABC16m12m?D25m? chapExoCorrec/11234 sacados/11234 BAC4m3mMN8m10m¸o˛o chapExoCorrec/8012 sacados/8012 ABC100o50oDEF50o30o
ABC34oDEF56o34o ROI55o112oDES55o13o ABCDE34o54o92o 32o41oABCDE107o OVSIR18o42o120o (dABCEF ABCH E.8013 Consider the two triangles ABC and DEF : 1 Show that the two triangles ABC and DEF are similar. 2 Complete the table below : Corresponding sides In triangle ABC [ AB ] [ BC ] In triangle DEF [ EF ] E.11254 Consider the two triangles DES and ROI : 1 Show that the two triangles DES and ROI are similar. 2 Complete the table below : Corresponding sides In triangle DES [ DE ] [ ES ] In triangle ROI [ RI ] E.8018 Consider the two segments [ CE ] and [ BD ] that intercept at A . Justify that the triangles ADE and ABC are similar trian-gles. E.8017 Consider a triangle ABC such that : ∠ DAE = 32 o ; ∠ BAC = 41 o and the points D and E belonging to the sides [ AB ] and [ AC ] respectively such that : ∠ ADE =107 o . Show that the triangles ABC and ADE are two similar tri-angles. E.11277 Consider triangle V OS and points I and R belonging respectively to segments [ OS ] and [ OV ] . Some angles have been indicated in the figure. 1 Justify that the two triangles V OS and ROI are similar. 2 List the three pairs of corresponding sides. E.8014 In the plane, consider three points A , B and C distinct. The straight lines ( AB ) and ( AC ) are intercepted by a straight line ( d ) , parallel to ( BC ) , at E and F , respectively. Justify that the triangles ABC and AEF are similar trian-gles. E.8016 Consider a triangle ABC rectangu-lar to C . We note H the foot of the height from the vertex C . Show that the triangles ABC and BHC are two similar tri-angles. https://chingmath.fr chapExoCorrec/8013 sacados/8013 ABC34oDEF56o34o chapExoCorrec/11254 sacados/11254 ROI55o112oDES55o13o chapExoCorrec/8018 sacados/8018 ABCDE34o54o92o chapExoCorrec/8017 sacados/8017 32o41oABCDE107o chapExoCorrec/11277 sacados/11277 OVSIR18o42o120o chapExoCorrec/8014 sacados/8014 (dABCEF chapExoCorrec/8016 sacados/8016 ABCH
ABCH ABCEFGH ABC34oDEF3cm1;8cm2;4cm56o ABC71oDEF3;7cm1;2cm3;5cm19o FED68oCAB26m24m10m22o 01234567891011121314151617ABC E.558 In the figure below, the triangle ABC is right-angled at C and H is the foot of the height from the vertex C . Show that the triangles ACH , CHB and ACB are similar triangles. E.10226 Consider the right triangle ABC with right angle at C shown below : Consider the square EFGH inscribed in triangle ABC such that side [ EF ] lies on the hypotenuse of the triangle. Justify that triangles ABC , AEH , BGF , and CHG are sim-ilar in pairs. 5. Similar triangles and the converse of the Pythagorean theorem E.11253 Consider the two triangles ABC and DEF : 1 Using the converse of the Pythagorean theorem, show that triangle DEF is a right triangle at F . 2 Justify that triangles ABC and DEF are similar trian-gles. E.11255 Consider the two triangles ABC and DEF : 1 Using the converse of the Pythagorean theorem, show that triangle DEF is a right triangle at F . 2 Justify that triangles ABC and DEF are similar trian-gles. E.11278 Consider the two triangles ABC and DEF : 1 Using the converse of the Pythagorean theorem, show that triangle ABC is a right triangle at B . 2 Justify that triangles ABC and DEF are similar trian-gles. 6. Similar triangles and CCC properties E.11246 Consider two points A (2) and B (7) on a graduated line and a third point C : https://chingmath.fr chapExoCorrec/558 sacados/558 ABCH chapExoCorrec/10226 sacados/10226 ABCEFGH chapExoCorrec/11253 sacados/11253 ABC34oDEF3cm1;8cm2;4cm56o chapExoCorrec/11255 sacados/11255 ABC71oDEF3;7cm1;2cm3;5cm19o chapExoCorrec/11278 sacados/11278 FED68oCAB26m24m10m22o chapExoCorrec/11246 sacados/11246 01234567891011121314151617ABC
POT31o24o125o32m40m64mEAU125o24o31o56m ABCMN2;4cm3cm3;6cm OUI4;2m¸o‚o˛oVAS7m8;5m5m¸o‚o˛o ABC36o2;4cmDEF3cm1;8cm54o36o UOI6m2;7m4;2m38o23oSEL38o119o4;5m 1 Place point I such that segment [ AI ] measures three times the length of segment [ AB ] . 2 a Draw the line ( d ) parallel to the line ( BC ) passing through the point I . b Label J as the point of intersection of lines ( d ) and ( AC ) . c Justify that angles ABC and AIJ are equal in mea-sure. d What can be said about triangles ABC and AIJ ? 3 Using your ruler, check that the lengths of triangle AIJ are three times those of triangle ABC . AB BC AC AI AJ IJ Proposition: if two triangles are similar, then their cor-responding sides have proportional lengths. Definition: the proportionality of the lengths of the sides between two similar triangles defines two coefficients of pro-portionality: the coefficient of enlargement and the co-efficient of reduction . E.11327 Consider the two triangles EAU and POT : 1 Justify that the two triangles are similar. 2 Determine the reduction coefficient between these two triangles. 3 a In triangle POT , give the side that is homologous to side [ AU ] . b Deduce the length of triangle [ AU ] . E.11247 In triangle ABC , lines ( MN ) and ( BC ) are parallel to each other. 1 Justify that triangles AMN and ABC are similar. 2 In triangle ABC , which side is similar to side [ MN ] ? Use this to find the ratio between the two similar triangles AMN and ABC . 3 In triangle ABC , which side is similar to side [ AM ] ? Use this to figure out its length. E.11243 Consider triangles OUI and VAS : These two triangles are similar (their angles are equal two by two) . 1 a In triangle OUI , which side is homologous to side [ AS ] of triangle VAS ? b Deduce the reduction coefficient from triangle VAS to triangle OUI . 2 a In triangle OUI , which side is similar to side [ SV ] of triangle VAS ? b Use this to figure out the length of segment [ OI ] . E.11193 L’exercice n’existe pas. E.8015 Consider the two triangles ABC and DEF : 1 Show that the two triangles ABC and DEF are similar. 2 Determine the length of segment [ AB ] . E.11279 Consider the two triangles OUI and SEL : 1 Justify that the two triangles are similar. 2 a In triangle SEL , give the side that is homologous to side [ OI ] of triangle OUI . b Determine the reduction factor from triangle OUI to triangle SEL . 3 Use this to find the length of side [ SE ] . https://chingmath.fr chapExoCorrec/11327 sacados/11327 POT31o24o125o32m40m64mEAU125o24o31o56m chapExoCorrec/11247 sacados/11247 ABCMN2;4cm3cm3;6cm chapExoCorrec/11243 sacados/11243 OUI4;2m¸o‚o˛oVAS7m8;5m5m¸o‚o˛o chapExoCorrec/11193 sacados/11193 chapExoCorrec/8015 sacados/8015 ABC36o2;4cmDEF3cm1;8cm54o36o chapExoCorrec/11279 sacados/11279 UOI6m2;7m4;2m38o23oSEL38o119o4;5m
SEL65o65o4mPOT50o4;8m AMIR¸o¸o˛o˛o‚o‚o9m4;2m7;5m ABC5;5mDE2m2;2m128o128o30o22o22o ABC34o1;2cmDEF3cm1;8cm2;4cm56o 88cm56cm20cmxABCDEF E.11252 Consider the two triangles SEL and POT : 1 Justify that triangles POT and SEL are similar. 2 Determine the reduction coefficient from triangle POT to triangle SEL . E.11244 Consider triangles AMI and AIR : The two triangles AMI and AIR are similar (equality of tri-angles two by two) . 1 a In triangle AIR , which side is homologous to side [ AM ] in triangle AMI ? b Deduce the magnification factor of triangle AIR rela-tive to triangle AMI . 2 a In triangle ARI , which side is homologous to side [ AI ] of triangle AMI ? b Deduce the length of side [ IR ] . E.11194 Consider the configuration below : 1 Justify that triangles ABC and AED are similar. 2 a Which side is homologous to side [ ED ] ? b Determine the coefficient of proportionality between these two triangles. 3 a Which side is similar to side [ AE ] ? b Deduce the length of side [ AB ] . E.11256 Consider the two triangles ABC and DEF : 1 Using the converse of the Pythagorean theorem, show that triangle DEF is a right triangle at F . 2 Justify that triangles ABC and DEF are similar trian-gles. 3 a Determine the reduction coefficient of triangle DEF to triangle ABC . b Deduce the three lengths of triangle ABC . 7. Similar triangles, CCC properties, rational coefficient E.11248 A player hits the cue ball off the top cushion () and sends it onto the opposite cushion () as shown below : We assume that : " when the ball hits the edge of the table, it rebounds at the same angle at which it arrived ". Determine how far the ball is from the pocket. Hint: Any attempt at a solution, even if incomplete, will be taken into account. E.565 ABCD is a square with center O and side 10 cm . The bisector of the angle BAC intersects the diagonal [ BD ] at K and the side [ BC ] at L . 1 Demonstrate that the triangles AOK and ABL are sim-ilar. 2 Calculate the reduction coefficient of the triangle ABL 8. Characteristic property CCC https://chingmath.fr chapExoCorrec/11252 sacados/11252 SEL65o65o4mPOT50o4;8m chapExoCorrec/11244 sacados/11244 AMIR¸o¸o˛o˛o‚o‚o9m4;2m7;5m chapExoCorrec/11194 sacados/11194 ABC5;5mDE2m2;2m128o128o30o22o22o chapExoCorrec/11256 sacados/11256 ABC34o1;2cmDEF3cm1;8cm2;4cm56o chapExoCorrec/11248 sacados/11248 88cm56cm20cmxABCDEF chapExoCorrec/565 sacados/565
ABC4m3m2mDEF5m3;75m2;5m 6m5;6m5mABC7;5m7m6;25mDEF51o60o69o 6m5;4m4;8mABC4m3;6m3;2mDEF49o59o72o 7m4.2m3.5mABC4m2.4m2mDEF59o49o72o 1ABC22o5m3mDEF22o8m4;8m 2GHI130o3;2m4;8mJKL130o4m6m E.11241 Proposition: If the sides of two triangles are proportional to each other, then these two triangles are similar. Consider the two triangles : 1 Complete the table below : The side longest " medium " shortest In triangle ABC In triangle DEF 2 Show that this table is a table of propositional logic. 3 What can we deduce about triangles ABC and DEF ? E.11249 Consider the two triangles below : 1 Prove that triangles ABC and DEF are similar trian-gles. 2 Give the measure of angle DEF . E.11250 Consider the two triangles below : 1 Prove that triangles ABC and DEF are similar trian-gles. 2 Give the measure of angle DEF . E.11251 Consider the two triangles below : 1 Prove that triangles ABC and DEF are similar trian-gles. 2 Give the measure of angle DEF . 9. Characteristic property: CAC E.11242 Proposition: If two triangles have an angle of the same measure and the sides adjacent to the angle are propor-tional, then these two triangles are similar. Among the pairs of triangles below, which ones form a pair of similar triangles : 10. Enlargement and reduction https://chingmath.fr chapExoCorrec/11241 sacados/11241 ABC4m3m2mDEF5m3;75m2;5m chapExoCorrec/11249 sacados/11249 6m5;6m5mABC7;5m7m6;25mDEF51o60o69o chapExoCorrec/11250 sacados/11250 6m5;4m4;8mABC4m3;6m3;2mDEF49o59o72o chapExoCorrec/11251 sacados/11251 7m4.2m3.5mABC4m2.4m2mDEF59o49o72o chapExoCorrec/11242 sacados/11242 1ABC22o5m3mDEF22o8m4;8m 2GHI130o3;2m4;8mJKL130o4m6m
KLM40o30o110o5cmDEF40o30o5cm3;4cm2;6cmXYZ30o5cm3;4cmUVW30o110o7cm4;76cmRST35o55o7cmGHI55o7cm4;2cm5;6cmNOP5cm3cm4cmABC35o6cm ABCH E.562 1 Justifying, determine all isometric triangles between them. Determine the missing measurements. 2 Locate similar triangles. Determine missing measurements. 11. Similar triangles: determination of the enlargement/reduction coefficient E.5251 The figure below is not to scale. Consider above a triangle ABC right-angled A such that ∠ ABC =30 o and AB =7 cm . H is the foot of the height from A . 1 Draw the figure full size on the copy. Leave the construc-tion lines visible on the copy. 2 Determine that : AH =3.5 cm 3 Show that the triangles ABC and HAC are similar. 4 Determine the reduction coefficient to go from triangle ABC to triangle HAC . https://chingmath.fr chapExoCorrec/562 sacados/562 KLM40o30o110o5cmDEF40o30o5cm3;4cm2;6cmXYZ30o5cm3;4cmUVW30o110o7cm4;76cmRST35o55o7cmGHI55o7cm4;2cm5;6cmNOP5cm3cm4cmABC35o6cm chapExoCorrec/5251 sacados/5251 ABCH
ABCDMNPFigure 1Figure 2 3;6cm40o51oABCD ABCMNKL E.9293 Note: In this exercise, results will be rounded to two deci-mal places if necessary. To build the set for a play (Figure 1) , Joanna has a rectangu-lar board ABCD measuring 4 m by 2 m from which she must cut out the three triangles for the set before stacking them on top of each other. She proposes cutting the plate as follows (Figure 2) . Triangle ADM satisfies the following conditions : Triangle ADM is a right triangle at A . AD = 2 m ADM = 60 o 1 Show that [ AM ] measures approximately 3.46 m . 2 The unused part of the plate is shown in a grid in figure 2 . Calculate the proportion of the plate that is not used, rounded to two decimal places. 3 To ensure that the triangles overlap smoothly, Joanna wants the three triangles AMD , PNM , and PDN to be similar. Show that this is indeed the case. 4 Joanna would like the magnification factor for going from triangle PDN to triangle AMD to be smaller than 1.5 . Is this the case? Justify your answer. 12. Areas E.1334 Consider the triangle ABC shown below : 1 a Draw the triangle DEF obtained by a factor 2 en-largement of the triangle ABC . b Check the proportionality between the side lengths of the two triangles ABC and DEF . 2 a Using the square, draw the heights from vertex C in triangle ABC and from vertex F in triangle DEF . b Give an approximate default value for the areas of tri-angles ABC and DEF . c What can be said about the comparison of these two areas? E.4774 Consider the two triangles AMN and ABC shown below où the lines ( MN ) and ( BC ) are par-allel: We note K the foot of the height of the triangle AMN from M . We denote L the foot of the triangle height ABC arising from B . The following measurements are given : AN =2.7 cm ; MN =1.5 cm ; BC =4 cm ; KM =0.9 cm 1 a Determine the measure of segment [ AC ] . b Justify the equality: AM AB = 3 8 . c triangle AMN is a reduction of triangle ABC . Give the associated reduction coefficient. 2 a Using question 1 c , determine the measurement of height [ BL ] . b Determine the areas A AMN and A ABC of the triangles AMN and ABC respectively. c Establish equality: A AMN A ABC = 9 64 3 What other reasoning would have found the result of question 2 c ? https://chingmath.fr chapExoCorrec/9293 sacados/9293 ABCDMNPFigure 1Figure 2 chapExoCorrec/1334 sacados/1334 3;6cm40o51oABCD chapExoCorrec/4774 sacados/4774 ABCMNKL
ABCHMN3;6cm1;2cm2;7cm7cm ABCEF2;4cm3;6cm3;2cm6;8cm 61oABCDEF3;2cm6cm6;8cm7;5cm8;5cm ABCEFD3cm6cm30o60o30o15o E.4775 Consider the triangle ABC où points M and N belong to the segments [ AB ] and [ AC ] , respectively, and the lines ( MN ) and ( BC ) are parallel to each other : 1 Determine the reduction factor of triangle AMN with respect to triangle ABC . 2 a Determine the area of the triangle ABC . b Deduce the area of the triangle AMN . E.4776 We consider the following configura-tion : 1 It is assumed that the triangle AMN is a reduction of the triangle ABC whose reduction factor is 2 3 . The tri-angle ABC having an area of 6.75 cm 2 . Give the area of the triangle AMN . 2 It is assumed that triangle AMN is a reduction of tri-angle ABC whose reduction factor is 3 5 . The triangle AMN having an area of 3.51 cm 2 . Give the area of the triangle ABC . E.4794 Consider the triangle ABC whose dimensions are: AB = 6.8 cm ; BC = 3.2 cm ; AC = 6 cm The point E belongs to the segment [ AC ] such that AE = 2.4 cm . The perpendicular to the line ( AC ) and passing through the point E intercepts the line ( AB ) at the point F . 1 Establish that the triangle ABC is a right triangle. 2 Determine the measure of segment [ EF ] . 3 It is accepted that triangle ABC is an enlargement of triangle AEF . Noting A ABC and A AEF the respective areas of triangles ABC and AEF , determine the value of the quotient : A ABC A AEF . 13. Unclassified exercises E.9292 The figure below is not shown at full size. The points C , B , and E are in line. The triangle ABC is right-angled A . The triangle BDC is right-angled at B . 1 Show that the length BD is equal to 4 cm . 2 Show that the triangles CDB and BFE are similar. 3 Sophie states that the angle ∠ BFE is a right angle. Is she correct? 4 Max states that the angle ∠ ACD is a right angle. Is he correct? E.1333 Consider the two triangles ABC and DEF shown below : Can we say that the triangle DEF is an enlargement of the triangle ABC ? https://chingmath.fr chapExoCorrec/4775 sacados/4775 ABCHMN3;6cm1;2cm2;7cm7cm chapExoCorrec/4776 sacados/4776 chapExoCorrec/4794 sacados/4794 ABCEF2;4cm3;6cm3;2cm6;8cm chapExoCorrec/9292 sacados/9292 61oABCDEF3;2cm6cm6;8cm7;5cm8;5cm chapExoCorrec/1333 sacados/1333 ABCEFD3cm6cm30o60o30o15o