Find area of the circumscribed circle of an isosceles triangle | Important Geometry skills explained

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Find area of the circumscribed circle of an isosceles triangle | Important Geometry skills explained

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[Ankitsingh-yjm]

Here, I always get most intresting questions, thankyou sir.

[harikatragadda]

Extend CP to meet the circle at D. Using the Intersecting Chord Theorem,
CP* PD = AP*PB
8*(2R-8) = 6*6
R= 25/4

[michaelkouzmin]

just another approach, may be more cumbersome:
1) Area of triangle ABC according to Heron is 48 sq.units;
2)Area of circumscribed triangle ABC = abc/(4r) => 48=10*10*12/(4*r) => r=300/48=25/4; ...

[charleswebb]

Thank you. I used the same first steps, but after working out the isosceles triangle, I continued taking the perpendicular all the way to the circumference, then used intersecting cords to work out the diameter ... etc.

[vara]

Every problem you solve is interesting. Makes me to view them compulsorily.

[lornawilliams]

Clear explanation. Thanks. Several methods possible. Can also apply trig using sine or cosine rule and angle subtended by chord at centre = twice angle at circumference.

[_iq]

We can also extend the CP so it hits the circle in a point and we will call it Z for example. Now we have this simple equation:
BP.AP=PC.PZ which is:
6×6=8×PZ
PZ=36÷8=4.5
now we have
2r=8+4.5=12.5
r=12.5÷2=6.25
finally the area: (if π=3.14)
3.14×(6.25)²=122.66

[santiagoarosam]

M es punto medio de AB → CMB es triángulo rectángulo tipo 3, 4, 5 → CM=8 → Potencia punto M respecto a la circunferencia: 6x6 = 8(2r-8) → r=100/16=6.25 → Área círculo = 6.25²(Pi)
Gracias y un saludo.

[luigipirandello]

Esse vídeo revela a beleza da matemática. Thank you Sir.

[Vredmusik]

Great video, please keep them coming :) The end result is 625/16*pi, but you wrote 125/16*pi on the last page.

[jonathanjose]

After finding cp, I used intersecting chords theorem, and r=6.25

[julieyeomans]

...roundabout way, but it works: Heron's Formula to find area of triangle and then plug area into A=1/2 bh. Solve for h. Use chord theorem to find the length of the rest of the diameter. Add the two parts to get diameter. Divide in two for radius and solve for the area of the circle.

[disgruntledtoons]

Base of the isoceles is 12, half of that is six. This means that the altitude of the triangle is 8. r^2 = (8 - r)^2 + 36. r^2 = 100 - 16r + r^2. r = 100/16 = 6.25. Area = pi * r^2 = 625 * pi/16.

[stephenbrand]

I used the cosine rule to find the cosine of angleACB, then found the cosine of 2xACB (using the cos 2A) rule. Angle AOB is twice angle ACB hence used the cosine rule again on triangle AOB to find r squared and so forth.

[TheMingilator]

I got the unknown length from the midpoint of ab to the circumference using rule of cords and added to the height of 8 to get the diameter and calculated the area from there

[Kame]

Another option is to use trigonometry to solve for r.

Since CO = AO, angles ✓CAO = ✓ACO:
✓ACO = ArcSin(6/10) ≈ 36.86989
✓AOC = 180-(36.86989 * 2) ≈ 106.26020

Then use the Sine Theorem:
r = (10 / Sin(106.26020)) * Sin(36.86989)
r = (125/12) * (3/5)
r = 25/4

[danieldanmola]

Wow..nice work..
Here is an alternative..since area of çircle=πr².. you can use the relation R=abc/4∆. ( This relation will always hold when an equilatera*l traingle is inscribed on a circle ) ..
Where a =10, b= 10 an c=12. And ∆ = area of an equilateral traingle which can be obtained by using the herons formula..
Hence R= 10×10×12/4(48)
R= 25/4..
Substitute into A=πr².
A =π(25/4)²( final answer 👌👌)😁😁.. either approach is okay!!!🔥🔥

[mahatmapodge]

Theres a fairly easy way to solve this once you know that ACP is a 3, 4, 5 triangle

The angles of ACP are

36.87°, 53.13°, 90°

So, the angle of the isosceles triangle ABC are
53.13°, 73.74°, 53.13°

That means the centre triangle of AOB has angles
16.26°, 147.48°, 16.26°

Therefore triangle AOP has angles
16.26°, 73.4°, 90°

We know the length of AP is 6 units and the angle (oap) is 16.26°

We know CP is 8 units

r = CP - OP

OP = AP tan(oap)

So the radius can then be found as
r = 8 - 6(tan16.26°)

Which is 6.2500232606

Giving an area of

122.7193764742 units


Thought it was an interesting way to solve it from known data

[АлександрГрошев-из]

Можно найти например cos<C, по теореме косинусов, потом sin<C, а потом найти R по теореме синусов.

[AlejandroCastilloRamirez]

i just got the bore methode of coordinate geometry with origing at A then all points are given get the perpendicular bisector to line AC subst x=6 and you have the radius :D