Geometry Puzzle | Find area of Yellow shaded semicircle inscribed in a right triangle

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Learn how to find the area of blue shaded circle inscribed in a right triangle. Important Geometry skills are also explained: area of the circle formula; Two-tangent theorem; Pythagorean Theorem. Step-by-step tutorial by PreMath.com
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Geometry Puzzle | Find area of Yellow shaded semicircle inscribed in a right triangle
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Пікірлер: 84

@theoyanto Жыл бұрын

Nice n easy , thanks again 🤓👍🏻

@PreMath Жыл бұрын

My pleasure 😊 So nice of you. Thank you! Cheers! 😀

@theoyanto Жыл бұрын

@@PreMathyour geometry problems help to keep me sane, they give me a welcome diversion.

@PreMath Жыл бұрын

@@theoyanto Keep it up, my dear friend. Stay blessed.

@batavuskoga 11 ай бұрын

I solvedd it with the second method. But when you have 12/16 = r/8, I always multiply diagonally. Same result, I know, but that's the easiest way for me. I love your way of explaining the math problems. Always a clear explanation, you are one of the best

@bigm383 Жыл бұрын

Wow, that semi circle thought he was some kind of tough guy!😂👍❤️

@PreMath Жыл бұрын

Well said! Thank you! Cheers! 😀

@JLvatron Ай бұрын

Tee-Hee!

@illyriumus2938 Жыл бұрын

OPC is a similar triangle with ABC and therefore, a 3-4-5 triangle: If PC=8 (4x2), the other side must 3x2=6.

@bb55555555 Жыл бұрын

yes I saw that too. radius had to be 6

@bb55555555 Жыл бұрын

also once you figured out that the bottom was 16 than that further confirms 6 because that last side of the triangle using the radius had to be 10

@phungpham1725 10 ай бұрын

This is what I did

@Abby-hi4sf Жыл бұрын

Thank you for showing the (AA ) similar triangles theorem use. Keep up the great work. You are great teacher

@PreMath Жыл бұрын

You're very welcome! So nice of you. Thank you! Cheers! 😀

@billcame6991 Жыл бұрын

My solution was similar to your second solution except I flipped OPC around the beta angle. BC is 16 and CP is 8 and both have the same angles. Therefore, OC is 10 and OP is 6. OP is the radius and I did half of the circle radius equation.

@amitavadasgupta6985 Жыл бұрын

Good presentation.

@maqboolahmad9301 Жыл бұрын

Keep it up

@PreMath Жыл бұрын

Thank you! Cheers! 😀

@XDXD-om9rc Жыл бұрын

Very nice explanations, thank you very much!

@PreMath Жыл бұрын

Glad you enjoyed it!

@KAvi_YA666 Жыл бұрын

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

@PreMath Жыл бұрын

So nice of you

@murdock5537 Жыл бұрын

Nice! tan⁡(φ) = 12/16 = 3/4 = r/8 → r = 6 → yellow area = 18π

@PreMath Жыл бұрын

Very good!

@quigonkenny 6 ай бұрын

By Two Tangent Theorem, PA=AB. Therefore CA = 8+12 = 20. As AB = 12 = 3(4) and CA = 20 = 5(4), this means that ∆ABC is a 4:1 ratio (3,4,5) Pythagorean Triple triangle and that BC = 4(4) = 16. As CA is tangent to Semicircle O at P, ∠CPO is 90°. As ∠BCA and ∠OCP are the same angle and ∠CPO and ∠ABC are both 90°, then ∠POC and ∠CAB are also the same angle and ∆ABC and ∆CPO are similar. BC/CP = AB/PO 16/8 = 12/r r = 12(1/2) = 6 A = (1/2)πr² = (π/2)6² = 36π/2 = 18π

@michaelkouzmin281 Жыл бұрын

The 3rd method as derivative of the the 2nd: tg(alpha) = 12/16; tg(alpha)= r/8; r/8 = 12/16; r= 8*12/16= 6; Ay = pi*r^2/2= pi*6^2/2=18*pi sq units.

@PreMath Жыл бұрын

Excellent! Thank you! Cheers! 😀

@kennethstevenson976 Жыл бұрын

I solved this problem using the Pythagorean Theorem, two triangles and two variables and arrived at 32r = 192; r=6 . Area = Pi x 6^2 /2 or 18Pi square units.

@martinwalker9386 Жыл бұрын

I computed using special triangles and similar triangles to find the radius.

@PreMath Жыл бұрын

Super!

@raya.pawley3563 11 ай бұрын

Thank you! If ABC is a 3-4-5 triangle (12, 16, 20), then so is OPC (6, 8, 10).

@phungpham1725 10 ай бұрын

I think this is the fastest way, kind of shortcut!

@spiderjump Жыл бұрын

AP = AB = 12 ( prop of tangent) AC = 12 + 8 = 20 by pythagorean theorem BC = SQROOT OF (20X20 -- 12X12) = 16 OC = 16 -- r where r is radius of semi circle by pyth theorem, (16 --r) sq = r sq + 8 x8 solve for r r = 6 hence semi circle area = pi36/2 = 18pi

@PreMath Жыл бұрын

Thank you! Cheers! 😀

@amit_ganjam Жыл бұрын

After getting BC=16, we can proceed as- ∆OPC ~ ∆ABC (By AAA similarity) Hence, OP/PC = AB/BC => r/8 =12/16 => r = 6 => A =πr^2/2 = 16π

@Copernicusfreud Жыл бұрын

Yay! I solved by the second method.

@PreMath Жыл бұрын

Bravo!

@samriddhadutta2104 Жыл бұрын

Please give some challenging questions from algebra trignometry . Menstruation is not included in our syllabus.

@wackojacko3962 Жыл бұрын

2nd method...now THAT'S GOD' SPEED! 😇

@PreMath Жыл бұрын

Enjoy! Thank you! Cheers! 😀

@Constantine817 Жыл бұрын

There is one more method: if BC=16, then 8²= 16*(16-2r); 64=256-32r; 32r=192; r=6; then Area= π*6²/2 = 18π

@PreMath Жыл бұрын

Excellent! Thank you! Cheers! 😀

@Constantine817 Жыл бұрын

@@PreMath My pleasure! Cheers!🙏

@marioalb9726 Жыл бұрын

Big Right triangle: Hypotenuse = 8 + 12 = 20cm h² = 20² -12² h = 16 cm Similarly of triangles: 16/20 = 8 / (h-r) h - r = 8 . 20 / 16 = 10 r = h - 10 = 6 cm Area = πr² /2 Area = 56,55 cm² ( Solved √ )

@misterenter-iz7rz Жыл бұрын

Let r be the radius, then OC^2=64+r^2, therefore 400-144=(r+sqrt(64+r^2))^2, 256=16^2=(r+sqrt(64+r^2)), (16-r)^2=64+r^2, r=192/32=6, then the answer is 6x6/2 pi =18 pi.

@MultiYesindeed Жыл бұрын

Love it.

@PreMath Жыл бұрын

Excellent! Thank you! Cheers! 😀

@giuseppemalaguti435 Жыл бұрын

Per costruzione lipotenusa é 8+12=20...per Pitagora risulta 20^2=12^2+(r+rad(64+r^2))^2....r^2=36....Ay=36pi/2=18pi

@ybodoN Жыл бұрын

Let's use BC as the axis of symmetry to reflect the diagram and get a circle inscribed in a triangle with sides a = 24 and b = c = 20. We can then use the general formula for the inradius of an incircle in a triangle: r = √[(s − a)(s − b)(s − c)/s] where s = ½ (a + b + c).

@sandanadurair5862 Жыл бұрын

Nice one

@philipkudrna5643 Жыл бұрын

B4 watching: The area is 18pi square units. The hypothenuse of the large triangle is 12+8=20. The bottom side is 16. (12^2+16^2=20^2 or 4 times the famous 3-4-5 right triangle). The radius r can be calculated as follows: r^2+8^2=(16-r)^2. This leads to r=6. The area of the semicircle is r^2*pi/2, which is 18pi.

@rishudubey1533 Жыл бұрын

thankyou v.much dear ❤

@PreMath Жыл бұрын

Most welcome 😊 So nice of you. Thank you! Cheers! 😀

@adampiechuta5774 Жыл бұрын

Solving for x could be faster if you noticed that tangens of angle C is 12/16 so is 6/8 and also r/8 so r=6.

@vidyadharjoshi5714 Жыл бұрын

Let OP = R. BC = 16. OC = 16 - R. In OPC OCsq = Rsq + 64. 256 - 32R + Rsq = Rsq + 64. 32R = 192. R = 6 Area = 18pi = 56.55

@PreMath Жыл бұрын

Excellent! Thank you! Cheers! 😀

@sumithpeiris8440 Жыл бұрын

If the semicircle cuts BC at D, then BD. BC = BP^2 So BD = 8^2/16 = 4 so BD = 16-4 =12 So radius = 6

@honestadministrator Жыл бұрын

∆POC is similar to ∆ BAC AB / BC = r / PC r = PC * AB / BC = 8 * 12 / √( 20^2 - 12^2) = 8 * 12 / 16 = 6 unit

@MrPaulc222 3 ай бұрын

I went for r^2 + 8^2 = (16-r)^2 for the radius, and went from there.

@devondevon4366 Жыл бұрын

18 pi or 56.55 Line AP= 12 (Tangent circle theorem)/ Hence AC = 20 (12+ 18) . BC = 16 since this is a 3-4-5 right triangle scaled up by 4. Draw a perpendicular line from P to O, forming a right triangle. This is the radius, r . Line OC = 16- r (since line BO =r and BC=16), and PC =8 (given). Using Pythagorean theorem: r^2 + 8^2 = (16-r)^2 r^2 + 64 = 256+r^2 - 32r 64-256 = - 32r -192 = - 32 r 6=r so radius of the semi circle = 6. The area of the circle is pi 6^2 or 36 pi, hence the area of the semi circle is 18 pi or 56. 55 Answer

@Aligakore Жыл бұрын

So, once again, I did not chose the easiest approach… Quickly figured that the hypothenuse of the right triangle ABC is 20 (12+8). I have decomposed BC as 2*r+x and OC as r+x. Noticed that ABC and OPC are similar triangles so (r+x)/r=20/12=5/3 => 1+(x/r)=5/3 => x/r=2/3 => x=2/3*r. BC can thus be written as 2*r+(2/3)*r=(8/3)*r. Using the Pythagorean theorem : ((8/3)*r)^2 + 12^2 = 20^2 => (64/9)*r^2=256 => r^2=36 => r=6 (a length cannot be negative).

@SAHIRVLOGCLP Жыл бұрын

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@AmirgabYT2185 3 ай бұрын

S=18π≈56,57

@amitsinghbhadoriya6318 Жыл бұрын

Thanks

@PreMath Жыл бұрын

You are very welcome! So nice of you. Thank you! Cheers! 😀

@ybodoN Жыл бұрын

Generalized: the area of the yellow semicircle is _π a² b / (2 (2a + b))_ where _a_ = AB and _b_ = PC.

@shrikrishnagokhale3557 Жыл бұрын

18π

@vara1499 Жыл бұрын

One method is a short step while the other is long one.

@PreMath Жыл бұрын

Excellent! Thank you! Cheers! 😀

@vara1499 Жыл бұрын

@@PreMath by the way, may I ask if you are of Asian origin ( specifically from india)?. Don't mistake me for asking this question.

@skipmars7979 Жыл бұрын

x-c-lent

@devondevon4366 Жыл бұрын

18 pi or 56.55

@militarymatters685 Жыл бұрын

18pi

@PreMath Жыл бұрын

. Thank you! Cheers! 😀

@JSSTyger Жыл бұрын

I think r = 6 and area = 18π

@PreMath Жыл бұрын

Thank you! Cheers! 😀

@DuaCreativeStudio Жыл бұрын

Only in 30 sec i got the answer

@prossvay8744 9 ай бұрын

Area of yellow shaded semicircle=18π

@DB-lg5sq 8 ай бұрын

r/8=12/16 r=6

@comdo777 Жыл бұрын

asnwer=18 is it hmm so than

@misterenter-iz7rz Жыл бұрын

Too early 😢

@PreMath Жыл бұрын

Thank you! Cheers! 😀

@shrikrishnagokhale3557 Жыл бұрын

18π