# Non Right Angled Triangles

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1 Non Right ngled Tringles Non Right ngled Tringles urriulum Redy

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3 Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit will show you how to use these trigonometri rtios to find the sizes of ngles nd length of sides. Try to nswer these questions now, efore working through the hpter. I used to think: How re sides leled in reltion to the leling ngles of non right ngled tringle? Drw nd lel tringle. The "osine rule" is used when two sides of the tringle nd the ngle etween them is given Wht is this rule? nswer these questions fter you hve worked through the hpter. ut now I think: How re sides leled in reltion to the leling ngles of non right ngled tringle? Drw nd lel tringle. The "osine rule" is used when two sides of the tringle nd the ngle etween them is given Wht is this rule? Wht do I know now tht I didn t know efore? 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

4 Non Right ngled Tringles sis Leling Tringles Everyone lels tringles the sme wy. Eh ngle is leled with pitl letter, nd their opposite sides re leled with the sme letter, ut non-pitl. For T Opposite to Opposite to Opposite to For TLMN Opposite to m Opposite to n M l N n m L Opposite to l K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

5 Non Right ngled Tringles Questions sis. Lel the sides in the following tringles. X Y J H K W. Lel the ngles in the following the tringles. m q p g f r 3. omplete the leling of the following tringles. T u M k s L 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

6 Non Right ngled Tringles Knowing More So fr we've used trigonometry in right-ngled tringles, ut we n tully use it to find sides nd ngles in ny tringle! The re Rule T is not right-ngled. The perpendiulr height, h, is drwn in. ` re # se # height re # # h We nsee: h sin ` h sin re # # sin h ` re sin So the re is hlf the produt of two sides nd the sine of their interior ngle. ny two sides nd their interior ngle will work: re sin or re sin or re sin Find the re of T DEF to deiml ple D 0 m 00 E F 4 m d nd f re given, so we will use re df sin E re TDEF dfsin E ^ 4 h^ 0 hsin f. 9. 7m ^deimlpleh Find the re of T to deiml ple nd re given, so we will use re re T sin sin (write this in T ) ^ h^ 8 hsin m ( deiml ple) 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

7 Non Right ngled Tringles Questions Knowing More. Find the re of the following tringles to deiml ple. 7 m m 0 m 6.4 m 78. T hs , ,. 4mnd. m. Drw the tringle. Find + to nerest degree. Find the re of the tringle to deiml ple. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 5 TOPI

8 Non Right ngled Tringles Questions Knowing More 3. n you find the res of the following shpes to the nerest squre m? E 0 m 3 6 m D 3 m Hint: Drw in Digonl D m 9.5 m m 0 m 93 6 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

9 Non Right ngled Tringles Questions Knowing More 4. Luke wnts to mke kite in the shpe elow. How mny squre m of mteril will he need? m m 44 D 5. The re of the tringle T PQR is 38.6 m. Find + Q. (Hint: Find+ P first) Q 0 m P m R 00% Non Right ngled Tringles K 8 7 Mthletis 00% 3P Lerning TOPI

10 Non Right ngled Tringles Using Our Knowledge The Sine Rule In ny tringle T we hve the formuls: sin sin sin or sin sin sin The proof is different depending on whether the tringle is ute or otuse. If T is n ute tringle If T is n otuse tringle h h D 80 - D In TD: h sin ` h sin In TD: h sin ` h sin In TD: ` sin sin h sin ` h sin (oth equl h) In TD: h sin^80 -h ` h sin^80 -h ` h sin Rememer sin^80 - θh sin θ ` sin sin ` sin sin (oth equl h) If we drew the ltitude from to then we ould show ` sin sin sin sin If we drew the ltitude from to then we ould show sin sin 8 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

11 Non Right ngled Tringles Using Our Knowledge Find the vlue of in T to deiml ples 0 m 8 7 m 8 Using the Sine Rule: ` sin sin 0 sin8 sin 8 ` 0 sin 8 sin 8 0^0469. fh f m f ( deiml ples) Find the vlue of + E in T DEF to the nerest degree D 36 E Using the Sine Rule: m F 3 m sine sin D e d ` sine sin 36 3 ` sin E 3 sin f - ` E sin ^0. 694fh f. 44 (nerest degree) Use the digrm elow to nswer these questions L Find + LPN to the nerest degree. N 55 0 m 3 9 m P 7 m M sin+ LPN sin + LNP LN LP ` sin+ LPN sin ` sin + LPN 0 sin ` + LPN sin ( ) ( nerest degree) Find the length of LM to the nerest m. First, find + LPM + LPM LPN Now, use the sine rule LM PM sin+ LPM sin + PLM ` LM 7 sin4 sin 3 ` LM 7sin m (nerest m) sin 3 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 9 TOPI

12 Non Right ngled Tringles Questions Using Our Knowledge The Sine Rule. Find the length y in the tringles elow to deiml ple. m y 6 37 y 9 m 4 0. Find the size of ngle in the following tringles: 9 m m 0 6 m 46 m 0 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

13 Non Right ngled Tringles Questions Using Our Knowledge 3. ship hs sils s drwn in the digrm elow: D 5 0 m 85 Find the length of. Find the lengths of D nd D. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

14 Non Right ngled Tringles Using Our Knowledge The osine Rule In ny tringle T we hve the formuls + - os or + - os or + - os The proof is different depending on if the tringle is ute or otuse. If T is n ute tringle If T is n otuse tringle h h x D (-x) D (x- ) x In T D: h + ^- xh h + x + -x (Pythgors) In T D: h + ^x- h h + x + -x (Pythgors) Susitute Susitute Susitute Susitute In T D: h + x (Pythgors) In T D: h + x (Pythgors) nd os x ` x os nd os x ` x os + - os + - os If the h is drwn from other orners we n show + - os or + - os If the h is drwn from other orners we n show + - os or + - os K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

15 Non Right ngled Tringles Using Our Knowledge Find the vlue of in T to deiml ples 8 m 30 5 m Using the osine Rule: ` ` + - os ^ h^8hos ^0. 866fh ` 9. 77f ` m ( deiml ples) Find the vlue of + Q to the nerest degree P 6 m 5 m Using the osine Rule: q p + r - pros Q ` ^0h^5hos Q R 0 m Q Mke os Q the sujet 0 ^ h^5hos Q os Q ` os Q f 300 Solve for + Q - + Q os ^ h 5. 56f. 6 (nerest degree) Find + in the following to deiml ple D 49. D + D - # D # D # os D ^ h^hos 49. This is the osine rule 5 m m f 9. 49fm Write this in the digrm + - # # # os 6 m 8 m ^9. 49fh ^6h^8 hos ^9. 49fh ` os 6 ^ h^8h f ` + os - ^0347. fh f ( deiml ple) 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

16 Non Right ngled Tringles Questions Using Our Knowledge The osine Rule 4. Find the length of the missing side x in eh of the following to deiml ple. D P 9 m x 5 m 73 x E m F Q 5 m 56 R 5. Find the vlue of the ngle in eh of the following to the nerest degree. 50 m 47 m 55 m 39 m W 6 m Y X 3 m 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

17 Non Right ngled Tringles Questions Using Our Knowledge 6. nyon hs the following dimensions: N 46 m L m M 50 m P To wlk round this nyon, would it e shorter to wlk long LNM or LPM? Use working to support your nswer. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 5 TOPI

18 Non Right ngled Tringles Questions Using Our Knowledge oth Rules 7. Use the sine nd osine rules to find the missing sides nd ngles in these tringles to deiml ples m 67 Q 38 m 55 m P 4 R 6 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

19 Non Right ngled Tringles Thinking More n miguous se For The Sine Rule Rememer tht sini sin(80-i). So when the sine rule is used to find n ngle (using sin - ) two nswers re possile. For exmple if sini 0.5 then i 30 or 50 (80-30). It is our jo to determine whether one or oth nswers re vlid. This depends on whether third ngle exists or not. Find + (to deiml ple) in T if + 50, 4m nd 7. m ording to the sine rule sin sin ` sin sin ` sin 7. # sin ` + sin ^094. h - ` or dd eh nswer to the given ngle. If this sum is less thn 80 then the nswer is vlid euse the third ngle exists. The given ngle in the question is + 50 ` ` vlid or ` vlid oth sums re less thn 80, nd so oth nswers for + re vlid. So T my look like either of these tringles: 4 m m 50 or 4 m m 50 se where oth nswers re vlid (like ove) is sid to e "miguous." 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 7 TOPI

20 Non Right ngled Tringles Thinking More Determining if nswers re Vlid Of the two possile nswers, one will lwys e otuse So if you know the tringle is ute, then the otuse ngle is invlid. If you know the missing ngle is otuse, then the ute ngle is invlid. (90to80 ) nd the other will lwys e ute ^0to 90h. The sum of ngles in tringle is 80. So if the sum of possile nswer nd the given ngle is 80 or more it mens tht the third ngle in the tringle is 0 or negtive. This is impossile, so the nswer would e invlid. Find + L (to deiml ple) in T LMN if + N 80, n m nd l m ording to the sine rule sinl sin N l n ` sinl sin ` sin L 40. 5# sin f ` + L sin ^0806. fh ` + L 55. 5f. 55. or +L dd eh nswer to the given ngle ^+ N 80h. ` + L+ + N ` vlid or + L+ + N ` invlid So only the first nswer is vlid nd + L 55.. This is not n miguous se. + L 55. nd T LMN looks like this M 40.5m 48.6m N L 8 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

21 Non Right ngled Tringles Questions Thinking More. In tringle T 9m, 5m nd + 0. Find two possile vlues for + to the nerest degree. hek if these nswers re vlid. Is this n miguous se? Drw rough sketh of the tringle(s). d Find the third ngle in the tringle(s) nd dd them to the digrm in. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 9 TOPI

22 Non Right ngled Tringles Questions Thinking More. In T PQR, + R 40, p 0.3m nd r Find the two possile vlues for + P to the nerest degree. Is this n miguous se (re oth ngles vlid)? Whih nswer is vlid if T PQR is ute? Whih nswer is invlid? d Whih nswer is vlid if + P is otuse? Whih nswer is invlid? 0 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

23 Non Right ngled Tringles Questions Thinking More 3. In tringle T JKL, + K 70, k 5. 4m nd l 4m. Find the vlid size(s) of + L. Find + J. Find the length of j. d Drw T JKL. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

24 Non Right ngled Tringles nswers sis: sis:. 3. X w Y T u y x s S t W H k j K h J U l M m k L. K Knowing More:. re T 8. 5m R re T 74. m q p. P F m r Q G m.m g f re T 89. 0m M K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

25 Non Right ngled Tringles nswers Knowing More: Thinking More: 3. Totl re m. + 6 or + 54 Totl re 3m oth nswers re vlid; therefore this is n miguous se. 4. Totl re 99m Q m 5m Using Our Knowledge: 0 6. y 9. m y. 7m 5m m 0 40 d + 34 or m(to d.p.) or + 0 D. 0.0m nd D. 6.0m oth nswers re vlid; therefore this is n miguous se. 4. x 9.6m (to d.p.) x.5m (tod.p.) If PQR is ute, the ngles re in the rnge 0-90, +P 70 is the vlid nswer nd + P 0 is invlid (tonerest degree) (tonerest degree) d If PQR is otuse, the ngles re in the rnge 90-80, +P 0 is the vlid nswer nd +P 70 is invlid. 6. It would e shorter to wlk long LPM thn LNM 3. + L 50 +J m (tod.p.) 9.89m (tod.p.) q 7.0m (tod.p.) d j 47. 3m K 70 4m 47.3m + Q (tod.p.) + R (tod.p.) J m 50 L 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

26 Non Right ngled Tringles Notes 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

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28 Non Right ngled Tringles

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