Calculate area of the right triangle | Green circle inscribed | Important Geometry skills explained

[@lindafromcalifornia1155]

You took a problem like that and broke it down beautifully. You do a very great service in the field of mathematics. You are a blessing.

[@PreMath]

Greatly appreciated😎 Thanks for your continued love and support! You are awesome, Linda. Keep it up 👍 Love and prayers from Arizona! 😀 Stay blessed 😀

[@lindafromcalifornia1155]

@@PreMath I've been blessed listening to you and so has countless others.

[@TurquoizeGoldscraper]

I started the same way you did, but I worked out the area as A = (1/2)(15 + r)(9 + r) 2A = 135 + 24r + r^2 Instead of trying to solve for r, I took the Pythagorean equation and substituted A like this: 306 + 48r + 2r^2 = 576 36 + 270 + 48r + 2r^2 = 576 36 + 2 * (135 + 24r + r^2) = 576 36 + 4A = 576 4A = 540 A = 135

[@satyanarayanmangal8900]

Instead of finding r We can directly put value of x²+24x Like Area = 1/2(9+x)(15+x) where x is FB=EB Therefore from Pythagoras theorem we get ( 9+x)² + (15+x)² = 24² x²+24x = 135 ...............(i) Now area = 1/2( 9+x) ( 15+x) =1/2 ( x² +24x +135) From (i) x²+24x = 135 So 1/2 ( 135 +135 ) 135 Sq units

[@prbmax]

Always so nice to see scaffolding in action with every minor step explained. Well done again.

[@PreMath]

Excellent! Thanks for your feedback! Cheers! You are awesome, Paul. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@selimbuyukkurt5618]

I think no need to find r. Area is (9+r) x (15+r) x 1/2= (135 + r^2 + 24r)/2. As 306 + 48 r + 2r^2 = 576. Simply 24 r+r^2=135. Back to area of triangle and replace r^2+24 r with 135. Final step (135+135)/2=135 . Thanks for exciting problems and clear explications

[@PreMath]

Thanks for your feedback! Cheers! You are awesome, Selim. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@vara1499]

You are systematic in teaching.

[@PreMath]

Glad you think so! Thanks for your continued love and support! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@murdock5537]

Nice and awesome, many thanks! sin⁡(φ) = (r + 9)/24 = k/24 cos⁡(φ) = (r + 15)/24 = p/24 → sin^2(φ) + cos^2(φ) = 1 → k^2 + p^2 = 24(24)→ r + 12 = √279 = 3√31 → r = 3(√31 - 4) → area ∆ = (9/2)(√31 + 1)(√31 - 1) = 135

[@kennethstevenson976]

I have to remember to read the original problem when I am giving the final answer. Since the circle was shaded I found the area of the circle which contains most of the work of the original problem instead of the area of the triangle, that was the easy part. I always correct my work when I make a mistake in the final steps of a problem, good lesson for me, always read the original problem before I submit the answer.

[@PreMath]

No worries! We are all human. I'm glad you did realize the mistake. Thanks for your feedback! Cheers! You are awesome, Kenneth. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@harrymatabal8448]

Mr kennith we will never learn if we don't make mistakes. My father made mistakes and I am one of them 😂😂😂

[@pwmiles56]

(15+r)^2 + (9+r)^2 = 24^2 [(15+r) - (9+r)]^2 = 24^2 - 2(15+r)(9+r) 6^2 = 24^2 - 4A 4A = 24^2 - 6^2 A = 12^2 - 3^2 = 135

[@AnonimityAssured]

I think we can use a little-known theorem to shorten this calculation a bit: Spoiler alert. If the point of contact between the hypotenuse and incircle of a right triangle divides the hypotenuse into lengths x and y, then the triangle has area x ∙ y. In this case, x = 15 and y = 9, so A = 15 ∙ 9 = 135.

[@beta700a]

I knew there should be an easier solution )) But I think the idea of PreMath is NOT to find the optimal solution, but quite opposite - to show the longest solution, so that on a course to the answer we can have an explanation about many math and geometry stuff - for educational purposes. You solution is absolutely perfect, but it has lesser educational value - just one theorem ☺ Anyway, thank you for sharing this little-known theorem! Math rules ))

[@ravisankar9669]

The opportunity of understanding the theorum,(tangential theorum),algebraic identies incorporated in solving the problem is important rather to solve the problem in short cuts.However these additional short cut methods displayed by some viewers are also heartily welcome as they help the learners to gain more knowledge in maths,especially to the beginners and more so to the Novice.

[@HappyFamilyOnline]

Very interesting👍 Thanks for sharing😊

[@ramanivenkata3161]

Your explanation is such that everyone can understand.

[@PreMath]

Glad to hear that! Thanks for your continued love and support! You are awesome, Ramani. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@hectorteano9348]

Thank you, Sir! I am learning a lot!

[@KAvi_YA666]

Thanks for video.Good luck sir!!!!!!!!!

[@PreMath]

So nice of you, dear Thanks for your continued love and support! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀

[@uwelinzbauer3973]

Hello professor premath, Same as Mr Kenneth Stevenson, I almost ended with calculating the area of the circle and forgot to calculate the area of the triangle. Perhaps I was confused, because the circle was shaded. That reminds us, always pay attention, always look carefully and always listen carefully... In any case: thanks for your straight and clear explanations of the way how to find the correct answers

[@PreMath]

No worries! We are all human. I'm glad you did realize the mistake. Thanks for your feedback! Cheers! You are awesome, Uwe. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@mateasevic4493]

😊Pp0 😊😊

[@alster724]

Amazing solution. I thought that no solution exists but it turns out there is

[@PreMath]

Thanks for your feedback! Cheers! You are awesome, Kevin. Keep it up 👍 Love and prayers from the USA! 😀

[@devondevon4366]

Draw a radius (90 degrees) from D to O and another from O to E; both are equal (Tangent Circle Theorem). That is, line A E =line AD = 15 Draw a radius from F to O; line CD= line CF= 9 EB=EF= radius of the circle since rectangle OFED is a square. The base of the triangle = 15 + r The height of the triangle =9 + r To find 'r.' c^2 =a^2 + b^2 = (15+9)^2 = (15 + f)^2 + (9+r)^2 [ Pythagorean Theorem ] 576 = 2r^2 + 48r + 306 288= r^2 +24 r + 153 divide both sides by 2 r^2 + 24 r -135=0 r = 4.7039; hence The base of triangle = [15 + 4.70329] = 19.70329 The height of the triangle = [9 +4.70329] = 13.70329 Area = [(19.70329)(13.70329)]/2 = 135 Answer

[@davidmorris4743]

BE=BF = x. Triangle ABC (Pythagoras) gives. x^2 + 24x - 135 = 0. Area o triangle ABC = 0.5(x^2 + 24x + 135) Substitute from 1st equation. Hence triangle = 135.

[@WaiWai-qv4wv]

Ok Very thanks

[@protradernb9703]

Make a video on stock market using trigonometry if u can...its may be helpfull for others

[@malcolmmcgrath9344]

I have a question regarding I think you would call it precision. We worked out the radius to be r = 3(√31 - 12) and went on to get a area of 135 sq units Suppose we said the radius was approx 4.7 and went on to get an area of approx 134.95 sq units After all when we are dealing with PI we use 3.14 . When is it right to right to round some number and to what precision 2 dec 4 dec ????

[@volodymyrzakolodyazhny]

IDK rules in all countries but generally we don't use approximate rounded values, but we use precise values only.

[@harrymatabal8448]

Excellent explanation premat but the question asked to find the area of the right triangle but I made a mistake and found the area of the wrong triangle. I don't know the difference between right, wrong and left. But I got the right answer.😂. Great video. Makes you think

[@protradernb9703]

How can we claculate degree factor of a hexagona. ( Wd gann). If 360 degree degree factor is 2 in circle and 1.75 in hexagon why? Especialy in hexagon 1.75 in which reason?

[@abhibagai5231]

there's no need to use the quadratic equations since r^2 + 24r=135 and The area of the triangle (15+r)(9+r)= r^+24r +135.... So after a quick substitution A= (135 + 135 )/2 = 135

[@SuperJdub99]

Any time a circle is circumscribed within a right triangle and you know the two lengths of the hypotenuse that touch the circle, those two lengths multiplied while always equal the area of the triangle. In this case 9 x 15 = 135, but it will work in any other example too, so long as the circle is circumscribed within the right triangle.

[@jayeshkumar3861]

No need to find r. Nice Q.

[@dezsodurst6965]

Let AB =a BC = b AC = c Radius of the circle = r AD = p DC = q D: tangent point ---------------- Given: p, q T find: A (Area of the triangle) ---------------------------------------- From TTT teorem: 1) a = p + r 2) b = q + r 3) c = p + q --------------- Dont' need r 1) - 2) a - b = p - q /^2 a^2 - 2ab + b^2 = (p - q)^2 a^2 + b^2 = c^2 c^2 - 2ab = (p + q)^2 (p + q)^2 -2ab = (p - q)^2 (p + q)^2 - (p - q)^2 = 2ab 4pq = 2ab /4 pq = ab/2 = A Area of the triangle A = p*q ========

[@vidyadharjoshi5714]

24sq = (15+r)sq + (9+r)sq. Solve for r. r = 4.7. Area = 0.5*(15+4.7)*(9+4.7) = 134.945

[@meandyou48364]

No video-torial this time?😅 But great videos....thank you so much!

[@spiderjump]

(9+r)^2 +(15+r)^2=24•24 Solve for r

[@devondevon4366]

Correct if he asks for the circle's radius, which is 4.70329. But he asked for the area of the triangle or ((15+ 4.70329)(9+ 4.70329))/2 = 135

[@davidangus1]

Is there a logical reason why the area is exactly the product of the two sections of the hypotenuse? I can't see it. If it is not a coincidence, then it is a very useful formula!

[@devondevon4366]

Good observation. Perhaps, but you could construct a different size circle inscribed in a right triangle, and then another, to draw your conclusion

[@theoyanto]

Nice job, one day I'll get some of these correct 😅

[@malcolmmcgrath9344]

I have exactly the same thoughts. I have managed two correct out of many * many!

[@PreMath]

Sure! Patience and perseverance pays off. Thanks for your feedback! Cheers! You are awesome, Ian. Keep it up 👍 Love and prayers from the USA! 😀 Stay blessed 😀

[@theoyanto]

@@PreMath my mum always told me the 5 P's Proper preparation prevents poor performance, she was soooo right 👍🏻

[@theoyanto]

@@malcolmmcgrath9344 just keep going 👍🏻

[@PreMath]

@@theoyanto Right on! Mom is always right 😀

[@davidellis1929]

One can determine the area of the triangle without needing to calculate the radius of the circle (which is irrational). Let r be the radius. Since OFBE is a square, FB=BE=r. By the Pythagorean theorem, (9+r)^2+(15+r)^2=(9+15)^2. If we subtract 9^2+15^2 from both sides, this yields 2r^2+48r=270. Don't bother to solve the quadratic equation. The area of the triangle is (1/2)(9+r)(15+r) = (1/2)(r^2+24r+135) = (1/4)(2r^2+48r+270) = (1/4)(270+270) = 135. In general, the area of the triangle is the product of the segments of the hypotenuse!

[@sorourhashemi3249]

I got it ignoring the circle. AE=15, cF=9, EB=a, BF=a, so (15+a)+(9+a)=24^2==>a=4.7 and area is134.967

[@helmsman13]

Fun fact: In this case the area of the triangle is always the product of the hypotenuse's parts.

[@D.P9]

Sir simply area of right angle triangle = 15×9 = 135

[@derhausfreund5691]

wiki A = 15 * 9 :-)

[@alexandermorozov2248]

Как можно геометрически представить случай, когда радиус отрицательный?

[@andirijal9033]

Area triangel ABC = 15 x 9 = 135, because this is teorema.

[@prossvay8744]

Area of the right triangle=135 square units

[@devondevon4366]

135

[@marimehlen2813]

Or 9*15= 135

[@davidguy9197]

un déliceà déguster

[@johnsports_iii]

Is it a coincidence that the area is 9 * 15?

[@philippedalleur5186]

It is exactly 9*15. If you put a=AD and b=DC, the areas of triangles AOE and AOD are the same and their total area is a.r, as well as COF and COD have a total area b.r. So the area S of ABC is a.r+b.r + r^2. And S is also (a+r).(b+r)/2 or 2S = a.r + b.r + r^2 + a.b = S + a.b. So S = a.b !😀

[@TxRealtorSA]

Takes me back to high school...1977. That's when kids really learned math instead of learning pronouns. Please start a math tutoring service so our students (that are being failed by the public school system) can move ahead in life.

[@PreMath]

Couldn't agree more! Our public school system is dumbing down... Thanks for your continued love and support! You are awesome. Keep it up 👍 Love and prayers from Arizona! 😀 Stay blessed 😀

[@abdulmenanseid6156]

8+15=23 not ever 24