青藜苑教育 www.thetopedu.com 010-6895997 1301951457 Revision Topic 9: Pythgors Theorem Pythgors Theorem Pythgors Theorem llows you to work out the length of sides in right-ngled tringle. c The side opposite the right-ngle is clled the hypotenuse it is the longest side in the tringle. If the sides re lbelled, b, c (with c being the length of the hypotenuse), then Pythgors theorem is b Exmple 1: Finding the length of the hypotenuse 8.5 cm x cm Find the vlue of x. 0.4 cm We strt by lbelling the sides, b, nd c: (c is the hypotenuse; nd b re the other two sides) c 8.5 cm x Write down Pythgors theorem: Put in the vlues of, b, c: 8.5 0.4 x Work out the left-hnd side: 7.5 + 416.16 = x Squre root: 488.41 = x x = 488.41 =.1 cm 0.4 cm b Finlly we should check the nswer seems resonble. The hypotenuse is the longest side in right-ngled tringle. s our nswer is bigger thn 0.4 cm, it seems resonble. Exmple : Finding the length of shorter side Q Find the length of PQ. 1.7 m b P 4.1 m c R We lbel the tringle, b, c (c must be the hypotenuse). Pythgors theorem: Substitute in the numbers: 1.7 4.1 Work out the squres:.89 16. 81 Subtrct.89 from both sides: 13. 9 Squre root: = 3.73 m ( deciml plces) Exmintion Question 1: 1 The nswer must be smller thn the hypotenuse
青藜苑教育 www.thetopedu.com 010-6895997 1301951457 The digrm shows the position of ferry siling between Folkestone nd lis. The ferry is t X. X 4km The position of the ferry from lis is given s: North of lis 15km, West of lis 4km. 15 km lculte the distnce of the ferry from lis. Give your nswer to one deciml plce. lis Exmintion Question F 6 cm In the digrm, tringle FGH is right-ngled t G. GH = 4cm nd FH = 6 cm. lculte the length of FG. G 4 cm H Pythgors Theorem in Isosceles Tringles n isosceles tringle cn be split into right-ngled tringles: It is therefore possible to use Pythgors theorem to find lengths in isosceles tringles. Exmple: Find the re of this tringle. 6cm Split the tringle into two: h 6 cm 9cm 4.5 cm We cn find the height of the tringle: h 4.5 6 h 0.5 36 h 15.75 So h = 3.97 cm 1 1 Therefore the re is h 9 3.97 17. 9 cm Exmintion Question 3 = 19.5 cm, = 19.5 cm nd = 16.4 cm.
青藜苑教育 www.thetopedu.com 010-6895997 1301951457 19.5 m D 16.4 m 19.5 m Note: Pythgors theorem could occur on the non-clcultor pper. Exmple for non-clcultor pper 4cm x Find the vlue of x. 5 cm Pythgors theorem: 4 5 x 16 5 x 41 x So, x = 41 cm Since you do not hve clcultor, leve the nswer s squre root. Do not try to estimte the squre root of 41 unless you re told to do so. Exmple : Non-clcultor pper 0 cm 11cm x cm Pythgors theorem: So, Find the vlue of x. 11 x 0 11 x x 9 x = 3 cm 0 Note tht 11 11 3
青藜苑教育 www.thetopedu.com 010-6895997 1301951457 ppliction of Pythgors Theorem: Finding the distnce between two points Exmple: The coordintes of the points nd re (6, 8) nd (1, 1). Work out the length of. Sketch (1,1) 5 units (6, 8) 7 units sketch showing the positions of points nd is n importnt first step. 7 5 49 5 74 = c So the length is 74 = 8.6 units Other uses of Pythgors Theorem Exmple 4 cm 7cm Prove tht the tringle is right-ngled. 5 cm Solution Pythgors theorem only works in right-ngled tringles, so we need to show tht the tringle bove stisfies Pythgors theorem. The longest side is 5 cm so this would hve to be the hypotenuse, c. The two shortest sides, nd b, re 7 cm nd 4 cm. Pythgors: Left-hnd side: 7 4 65 Right-hnd side: c 5 65 So the right-hnd side is equl to the left-hnd side, so Pythgors works in this tringle. Therefore the tringle is right-ngled. Specil tringles - right-ngled tringles with whole number sides The tringle t the top of this pge is specil s it is right-ngled tringle tht hs sides which re ll whole numbers. Two other common right-ngled tringles with sides tht re whole numbers re: This is clled 3cm 5cm the 3, 4, 5 tringle 5cm 13 cm 4cm 1 cm You cn use these bsic tringles to get other right-ngled tringles by multiplying ll the sides by the sme number. For exmple, tringle with lengths 6cm, 8cm nd 10cm would be right-ngled 4 (s its sides re double those in the 3, 4, 5 tringle).
青藜苑教育 www.thetopedu.com 010-6895997 1301951457 Right-ngled tringles in semi-circles Look t the tringle shown here. If is dimeter of the circle, then tringle hs right-ngle t (this is one of the circle theorems n ngle in semi-circle is 90 degrees). Exmple The side is the dimeter of circle. Find the length mrked. Give your nswer to 1 deciml plce. 1cm 16cm Solution s is dimeter of circle, ngle is right ngle. Therefore the hypotenuse is 16 cm. Pythgors: 1 16 144 56 56 144 11 = 11 = 10.6 cm (to 1 dp) Further exmintion question Find length. Give your nswer correct to 3 significnt figures. 5m 10m (Hint: You will need to find length D first, using the lower tringle). 8m D 5