Thank you so much Hanafy for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃 Love and prayers from the USA! Jazakallah😀
@@PreMath I am a teacher of English in Saudi Arabia. I Like math very much. I always visit your channel. I am always impressed by your methods of solution.
@@Osama-js2gw I'm sure you are also an awesome teacher. Thank you so much my friend for your continued love and support. Take care dear and stay blessed😃
You're very welcome Philip! Thanks dear for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃
I know the formulae for sine and cosine rules, but wanted to solve this one just using Eucledian geometry,and without any trigonometric formulae.Glad that I did it using eucledian geometry only.Wonderful pblm..thanks for sharing.
Well done! Thanks Jerry for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃
Wow - Instead of taking the sqrt of BC in the first calculation, then squaring again in the second calculation, you just left as sqrt of 49. I never would have thought of that! :)
Very nice method. Another way of doing this is to work on triangle PBC. Using cosine rule gives ||CB|=7.Then using sine rule find measure of angle CBP That gives the remaining two angles in triangle APB. Using sine rule in triangle APB gives x.Then area of green triangle equals half of x by 3 by sine of 120 degrees.
An alternative method is to use analytical geometry: find the coordinates of Point D by the intersection of Circles x^2 + y^2 = 9 and x^2 + (y-7)^2 = 25. You obtain D(15*sqrt(3)/14, 33/14). Then the height of the triangle is 33/14 and the base can be found using tan(120-theta) where tan theta = Dx/Dy=(15*sqrt(3)/14)/(33/14). Now AB= Dx + tan(120-theta) * Dy which leads to AB=18.19 and therefore the required area A=1/2 * AB * Dy=21.43.
When you have BC you can find angle PBC then sin of PBA then area using the formula of the triangle's area, with one side and 3 angles. (With a calculator it is even easier).
There should have been a comment that this is a one dimensional figure. The drawing makes it look 3 dimensional! I would never have thought the third angle was 120 degrees.
Use the Cosine Rule for triangle BPC, CB^2=CP^2+PB^2-2*CP*BP cos120 = 5^2+3^2-2*5*3 cos 120 =25+9-30*(-1/2) =25+9+15=49 CB= sqrt(49)=7. Use the Cosine Rule again on triangle BPC CP^2=PB^2+CB^2-2*PB*CB*cos PBC 5^2 =3^2+7^2-2*3*7*cos PBC 25 =9+49-42 cos PBC -33 = -42 cos PBC cos PBC = 11/14 Height of Green Triangle APB =PB*cosPB3*(11/14) =3*(11/14)=33/14 Now,consider triangle APB, Angle PAB=180-120-arcsine(11/14)=60-arcsine(11/14) =60-51.7867893=8.2132107 We now use the Sine Rule, PB/sin(8.2132107)=AB/sin(120) AB=PB*sin(120)/sin(8.2132107) =3*sqrt(3)/(2*0.1428571428)=18.18640154 Area of green triangle APB=(AB*Ht. green triangle)/2 ={(18.18640154*(33/14)}/2 =(18.18640154*(33/28) =21.43397324 square units. and that is your answer. Assisted by calculator. This is a good place to stop.
هذا الشكل الهندسي والمطلوب إيجادها مساحة المنطقه الخضراء وحسب اليابان التي أجرتها قبل عدة أيام فيجب علينا إيجاد قبل الزاويه bap والتي مفارقتها حسابيا ضلع القاعدة بالزاويه 120 =٣ سم وهو يساوي تماما بالزاويه 90=2.77سم اي انه نقص .223 ملم على طول 1.321 سم فإن الحسابات تكون كمل يلي: الجزءالاول.2×1.2÷2+الجزء الثانيوحساباته كأآتي1.321×(30÷2.23)_1.321×2.77÷2 +2.67×1.2÷2=مساحة المنطقه الخضراءسم
Wonderful as usual
Thank you so much Hanafy for taking the time to leave this comment. You are awesome 👍
Take care dear and stay blessed😃
Love and prayers from the USA!
Jazakallah😀
@@PreMath I am a teacher of English in Saudi Arabia. I Like math very much. I always visit your channel. I am always impressed by your methods of solution.
@@Osama-js2gw
I'm sure you are also an awesome teacher. Thank you so much my friend for your continued love and support. Take care dear and stay blessed😃
Nice to feature Al-Kashi‘s law of cosines and the sine-Formula for the area of a triangle! Thank you!
You're very welcome Philip!
Thanks dear for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃
Been MANY years since I studied that but you helped me remember the sine and cosine formulas for non right triangles. Thank you!
Thanks Fred for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
This is the coolest Channel I have seen
Thanks Syed dear for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
@@PreMath I will stay blessed thanks 😊😊😌 I feel relaxed 😌😊😊
I know the formulae for sine and cosine rules, but wanted to solve this one just using Eucledian geometry,and without any trigonometric formulae.Glad that I did it using eucledian geometry only.Wonderful pblm..thanks for sharing.
Well done!
Thanks Jerry for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃
Wow - Instead of taking the sqrt of BC in the first calculation, then squaring again in the second calculation, you just left as sqrt of 49. I never would have thought of that! :)
Thanks Mark for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
Very nice method. Another way of doing this is to work on triangle PBC. Using cosine rule gives ||CB|=7.Then using sine rule find measure of angle CBP That gives the remaining two angles in triangle APB. Using sine rule in triangle APB gives x.Then area of green triangle equals half of x by 3 by sine of 120 degrees.
Great tip!
Thanks John for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
S=99√3/8≈21,435≈21,44 🔥
An alternative method is to use analytical geometry: find the coordinates of Point D by the intersection of Circles x^2 + y^2 = 9 and x^2 + (y-7)^2 = 25. You obtain D(15*sqrt(3)/14, 33/14). Then the height of the triangle is 33/14 and the base can be found using tan(120-theta) where tan theta = Dx/Dy=(15*sqrt(3)/14)/(33/14). Now AB= Dx + tan(120-theta) * Dy which leads to AB=18.19 and therefore the required area A=1/2 * AB * Dy=21.43.
Pretty good sir!
Thanks for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
Stay safe sir! Thanks anyway sir
8:50 can some one explain?
I learned that A=a×b× sin x ,x
You are so awesome 😎😎😎😎
You are the best Syed!
@@PreMath thanks
When you have BC you can find angle PBC then sin of PBA then area using the formula of the triangle's area, with one side and 3 angles. (With a calculator it is even easier).
Thanks dear for the feedback. You are awesome.
Cheers
@@PreMath you are more awesome 😎💯😎 enjoy your life
Elegant solution
There should have been a comment that this is a one dimensional figure. The drawing makes it look 3 dimensional! I would never have thought the third angle was 120 degrees.
Thank you sir
Thanks for the visit! You are awesome Gowri👍 Take care dear and stay blessed😃 Kind regard
@@PreMath you are awesome so much and the coolest Channel I have seen
Use the Cosine Rule for triangle BPC,
CB^2=CP^2+PB^2-2*CP*BP cos120
= 5^2+3^2-2*5*3 cos 120
=25+9-30*(-1/2)
=25+9+15=49
CB= sqrt(49)=7.
Use the Cosine Rule again on triangle BPC
CP^2=PB^2+CB^2-2*PB*CB*cos PBC
5^2 =3^2+7^2-2*3*7*cos PBC
25 =9+49-42 cos PBC
-33 = -42 cos PBC
cos PBC = 11/14
Height of Green Triangle APB =PB*cosPB3*(11/14)
=3*(11/14)=33/14
Now,consider triangle APB,
Angle PAB=180-120-arcsine(11/14)=60-arcsine(11/14)
=60-51.7867893=8.2132107
We now use the Sine Rule,
PB/sin(8.2132107)=AB/sin(120)
AB=PB*sin(120)/sin(8.2132107)
=3*sqrt(3)/(2*0.1428571428)=18.18640154
Area of green triangle APB=(AB*Ht. green triangle)/2
={(18.18640154*(33/14)}/2
=(18.18640154*(33/28)
=21.43397324 square units. and that is your answer.
Assisted by calculator.
This is a good place to stop.
Wow 😍
Thanks Sravani for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
هذا الشكل الهندسي والمطلوب إيجادها مساحة المنطقه الخضراء وحسب اليابان التي أجرتها قبل عدة أيام فيجب علينا إيجاد قبل الزاويه bap والتي مفارقتها حسابيا ضلع القاعدة بالزاويه 120 =٣ سم وهو يساوي تماما بالزاويه 90=2.77سم اي انه نقص
.223 ملم على طول 1.321 سم
فإن الحسابات تكون كمل يلي:
الجزءالاول.2×1.2÷2+الجزء الثانيوحساباته كأآتي1.321×(30÷2.23)_1.321×2.77÷2
+2.67×1.2÷2=مساحة المنطقه الخضراءسم
#Cosine #Trigonometry
Cosine 120°
#LawofCosines
cos 120° = -0.5
Trigonometry
Cosine