Let A(0,0), then B(6,0), C(6,-13), D(0,-18) O is the centre of the circle, then O(k,-13/2) OC=OD (k-6)^2+(13/2)^2=k^2+(23/2)^2 k=-9/2 radius=OC=sqrt((-9/2-6)^2+(13/2)^2)=sqrt(610)/2
15:04 Once you've found PA, you can you can directly find PC using Pythagoras theorem (PBC is a right angled triangle). PC is the diameter as it projects a right angle on the circumference at B. PC /2 is the radius.
I extended the 18 line to meet the circle, giving a line length 23 (symmetry). I constructed the perpendicular bisector of the 13 chord (which must pass through the centre of the circle and bisect the line length 23). If left intersect of perp to line 18 or 23 is "a" and from there to right intersect is "b" then we have two equations in a,b using the two chords length 23 and 13: a.b=23²/4 and (a+6)(b-6)=13²/4 which can be solved by substituting for b in the second eqn using the first eqn. Then a+b=2r so r=(a+b)/2
Start off as you did in method 2 up to finding lengths of chords (14:58). At this point, draw perpendicular from O to BP at point Q. Quadrilateral OQBN is then a rectangle with OQ = BN = 13/2 (since ON is perpendicular to BC, it bisects chord BC) ON = BQ = 21/2 (since OQ is perpendicular to BP, it bisects chord BP) Diameter of rectangle = OB = r OB² = OQ² + BQ² r² = (13/2)² + (21/2)² = 169/4 + 441/4 = 610/4 r = (√610)/2
Center at origin and B = (x,y) => x^^2 + y^^2 = r^^2 (x-6)^2 + (y-18)^2 = r^2 and x^2 + (y-13)^2 = r^2. Subtract the first to the 2 others and we get 2 linear equations with x and y. Solving gives x = 21/2 and y = 13/2 and this gives r.
Another way to do it is to draw the segment lines from O to D and B which are the radius of the circle. name x the distance from the center O to M. We consider the right triangles ONB and OMD. Observe that the length of the side BN is 13/2 and the length of MD is 18-13/2=23/2. Now use Pythagoras theorem for both triangle to write: R^2=(x+6)^2+(13/2)^2 and R^2=x^2+(23/2)^2. From this point no more geometry, now only algebra. It follows that (x+6)^2+(13/2)^2=x^2+(23/2)^2. This equation simplifies itself into 12x-54=0 or x=9/2. Therefore, R^2=(9/2)^2+(23/2)^2=81/4+529/4=610/4. Finally, R=sqrt(610)/2.
Extending BA to the left side of the circumference and a parallel chord from C also to the left circumference, we see there are 5 units below 13 units in the extended AD chord is 23 units, and the extended chord of AB is 21 units long. Center of the circle is at (21/2, 23/2) = (10.5, 11.5). Now R^2 - (11.5) ^2 + (10.5 - 6) ^2 = 11.5^2 + 4.5^2 = [23^2+9^2]/4 = (529+81)/4 = 610/4 --> R=sqrt (610) /2.
Let D=(0,0), then C=(6,5), B= (6,18) Let O be the center. Since the horizontal line from passing O bisects BC, we know O=(x, 11.5) Let M be the midpoint of C and D, we know M=(3, 2.5) Since OM is perpendicular to CD, we have (x-3,9) inner product with (6, 5)=0 6x-18+45 = 0, hence x=-4.5 Therefore, the radius r=\sqrt{(-4.5)^2+11.5^2}=(1/2)\sqrt{9^2+23^2}=sqrt{610}/2
Las alineaciones BA y DA cortan la circunferencia en los puntos F y E respectivamente→ Potencia del punto A respecto a la circunferencia = AB*AF=AD*AE→ 6a=18*(18-13)→ a=15 → FC²=BF²+BC²→ (2r)²=(6+a)²+13²=21²+13²→ r=(√610)/2 =12,34908.... Gracias y un saludo cordial.
Give coordinates to the different points: A(u-6,v), B(u,v), C(u,v-13) and D(u-6,v-18). Since B, C and D are on a circle with radius r you get 3 equations with 3 unknowns u,v,r. The equations are: u^2+v^2=r^2; u^2+(v-13)^2=r^2; (u-6)^2+(v-18)^2=r^2. Out of the first 2 equations you get v= 13/2 and using the third equation you get u=21/2. Using then the first equation you get r = sqrt(610)/2.
I think it’s much simpler to use the formula for a circle. (x-a)^2 + (y-b)^2 = r^2 where a and b are the center coordinates of the circle. The intersection points are (0,0),(6,5)and(6,18) (arbitrary origin on bottom point). Then solve the three equations generated by the three intersection points. Easy!
By symmetry, AF = 5, and by chords theorem AP= 15 and AB =6 given. Then form ∆ PFB, the radius is the product of diagonal ( PF * FB) divided by the double the height of ∆ (2*5) Hence,( √(15^2+5^2) * √(6^2+5^2))/(2*5) √250*√61/10 = 5*√610/10 = √610/2
Let O be the center of the circle. Draw radius OT, so that OT is perpendicular to BC and intersects it at M. As OT is perpendicular to BC and BC is parallel to AD, OT is also perpendicular to AD, and intersects it at N. As BC is a chord, and as OT is an intersecting radius that is perpendicular to BC, then OT bisects BC, thus BM = MC = 13/2. As ∠ANM = 90° and ∠NMB = 90°, then ABMN is a rectangle, and thus NM = AB = 6 and AN = BM = 13/2. As AD = 18, ND = 18-13/2 = 23/2. Extend AD from A to E so that ED is a chord. OT bisects ED for the same reasons as it does BC, therefore EN = ND. EN = ND EA + AN = ND EA + 13/2 = 23/2 EA = 10/2 = 5 By the intersecting chords theorem, if two chords of a circle intersect, then the products of the lengths of each side of the intersection are equal for each chord. Extend AB from A to F, so that FB is a chord. As A is the intersection point between FB and ED, then FA•AB = EA•AD. FA•AB = EA•AD 6FA = 5(18) = 90 FA = 90/6 = 15 Method 1: Draw OB, and draw radius OS such that OS is perpendicular to FB and intersects it at P. As above, OS bisects FB, so FP = PB = (15+6)/2 = 21/2. As PB is perpendicular to BC, PB = OM = 21/2. As BM is perpendicular to FB, OP = BM = 13/2. Triangle ∆OBM: BM² + OM² = OB² (13/2)² + (21/2)² = r² r² = 169/4 + 441/4 = 610/4 r = √(610/4) = √610/2 Method 2: If two chords intersect perpendicularly, then the sum of the squares of the divided chord segments equals 4 times the radius squared. AB² + FA² + AD² + EA² = 4r² 6² + 15² + 18² + 5² = 4r² 4r² = 36 + 225 + 324 + 25 = 610 r² = 610/4 r = √(610/4) = √610/2
If I understand these methods, the first method triangulates to find the radius using the Pythagorean Thrm twice then uses a circle theorem and the second method requires applying chords twice. I thnk that this is the first time chords are diffcult to visualize.
Alternative "Insane" Method For number-crunching lovers only. If you're at all error prone with long complicated equations... don't even bother. If you like spending at least half an hour doing tons of algebra on a single problem... go for it. It took me three attempts, but it does work. Use chord bisector radius to create two Pythagoras equations. r² = (r - x)² + (13/2)² r² = [r - (x + 6)]² + [18 - (13/2)]² Have fun!
At 13:39 PBC is an inscribed rectangle triangle then CP is a diameter; from 14:58 all you need is to use Pythagoras to find |CP|, and then R=|CP|/2: |CP|=√(21²+13²)=√(610), R= √(610)/2.
Why not just use triangles OFM and OBN? We will have r^2=OM^2+MF^2 and r^2=ON^2+NB^2. NB=13/2=6.5, MF=MD=18-6.5=11.5, ON=OM+6. Let OM=x, then r^2=x^2+11.5^2, r^2=(x+6)^2+6.5^2. Subtracting first from second: 0=(x+6)^2+6.5^2-x^2-11.5^2=2*x*6+6^2+6.5^2-11.5^2;x=((11.5-6.5)(11.5+6.5)-36)/12=(5*18-36)/12=54/12=27/6. Then r=sqrt(27^2/6^2+11.5^2)=sqrt(152.5)=12.349.....
Much faster would be following. If distance from the center to AD we lebel x, than: R^2=x^2 + 11.5^2 and R^2=6.5^2 + (6+x)^2 Two equations and two unknowns.
Why not use Analytic Geometry and intersect 2 cords to get the center of the circle? The 2 lines are defined by perpendicular by-sectors of the 2 cords. It looks these RUclips math teachers hate Analytic Geometry.
I was following your few videos, your solution is good, but you need to improve a bit. That is you first explain the problem, then explain to get the desired result what are the information is needed, and how can get those information. Then you start deriving the solution.
The solutions offered lack elegance. I much rather have my teeth extracted (without anesthetics) than attack this problem. What was the motivation in offering this problem?
Let A(0,0), then B(6,0), C(6,-13), D(0,-18)
O is the centre of the circle, then O(k,-13/2)
OC=OD
(k-6)^2+(13/2)^2=k^2+(23/2)^2
k=-9/2
radius=OC=sqrt((-9/2-6)^2+(13/2)^2)=sqrt(610)/2
15:04 Once you've found PA, you can you can directly find PC using Pythagoras theorem (PBC is a right angled triangle). PC is the diameter as it projects a right angle on the circumference at B. PC /2 is the radius.
I extended the 18 line to meet the circle, giving a line length 23 (symmetry). I constructed the perpendicular bisector of the 13 chord (which must pass through the centre of the circle and bisect the line length 23).
If left intersect of perp to line 18 or 23 is "a" and from there to right intersect is "b" then we have two equations in a,b using the two chords length 23 and 13:
a.b=23²/4 and (a+6)(b-6)=13²/4 which can be solved by substituting for b in the second eqn using the first eqn.
Then a+b=2r so r=(a+b)/2
Start off as you did in method 2 up to finding lengths of chords (14:58).
At this point, draw perpendicular from O to BP at point Q. Quadrilateral OQBN is then a rectangle with
OQ = BN = 13/2 (since ON is perpendicular to BC, it bisects chord BC)
ON = BQ = 21/2 (since OQ is perpendicular to BP, it bisects chord BP)
Diameter of rectangle = OB = r
OB² = OQ² + BQ²
r² = (13/2)² + (21/2)² = 169/4 + 441/4 = 610/4
r = (√610)/2
Center at origin and B = (x,y) =>
x^^2 + y^^2 = r^^2
(x-6)^2 + (y-18)^2 = r^2 and
x^2 + (y-13)^2 = r^2. Subtract the first to the 2 others and we get 2 linear equations with x and y. Solving gives x = 21/2 and y = 13/2 and this gives r.
You found the area of the triangle BCD wrong as 39. Hight of that triagle cannot be 6
it is 39 man.just analyze,if u cant,then use software
Height of this triangle is outside the triangle, so it is 6. You can also use Heron's formula to compute area and you will get 39.
Another way to do it is to draw the segment lines from O to D and B which are the radius of the circle. name x the distance from the center O to M. We consider the right triangles ONB and OMD. Observe that the length of the side BN is 13/2 and the length of MD is 18-13/2=23/2. Now use Pythagoras theorem for both triangle to write: R^2=(x+6)^2+(13/2)^2 and R^2=x^2+(23/2)^2. From this point no more geometry, now only algebra. It follows that (x+6)^2+(13/2)^2=x^2+(23/2)^2. This equation simplifies itself into 12x-54=0 or x=9/2. Therefore, R^2=(9/2)^2+(23/2)^2=81/4+529/4=610/4. Finally, R=sqrt(610)/2.
same here
so do i
👍👍👍
Extending BA to the left side of the circumference and a parallel chord from C also to the left circumference, we see there are 5 units below 13 units in the extended AD chord is 23 units, and the extended chord of AB is 21 units long. Center of the circle is at (21/2, 23/2) = (10.5, 11.5). Now R^2 - (11.5) ^2 + (10.5 - 6) ^2 = 11.5^2 + 4.5^2 = [23^2+9^2]/4 = (529+81)/4 = 610/4 -->
R=sqrt (610) /2.
Let D=(0,0), then C=(6,5), B= (6,18)
Let O be the center. Since the horizontal line from passing O bisects BC, we know O=(x, 11.5)
Let M be the midpoint of C and D, we know M=(3, 2.5)
Since OM is perpendicular to CD, we have (x-3,9) inner product with (6, 5)=0
6x-18+45 = 0, hence x=-4.5
Therefore, the radius r=\sqrt{(-4.5)^2+11.5^2}=(1/2)\sqrt{9^2+23^2}=sqrt{610}/2
延长BA 交于○上F,连接FC ,得到方程如下：设AF为X,直径为D ，FD为Y
(X+6)²+13²=D²
Y²=D²+(√61)²
x²+18²=Y²
计算得到X=15,代入 得到D为√610，R等1/2*√610
r is the radius of the BCD triangle's circumcircle: r=abc/4T[BCD](triangle)
a=BC=13
b=BD=SQRT(AB^2+AD^2)=18,9737
c=CD=SQRT(CE^2+DE^2)=7,8103
T[BCD](triangle)= T[ABCD](trapezoid)-T[ABD](triangle)
T[ABCD](trapezoid)= (AD+BC)/2*AB= 93
T[ABD](triangle)= AB*AD/2= 54
T[BCD](triangle)= 39
r= 13*18,9737*7,8103/(4*39)= 12,349
Las alineaciones BA y DA cortan la circunferencia en los puntos F y E respectivamente→ Potencia del punto A respecto a la circunferencia = AB*AF=AD*AE→ 6a=18*(18-13)→ a=15 → FC²=BF²+BC²→ (2r)²=(6+a)²+13²=21²+13²→ r=(√610)/2 =12,34908....
Gracias y un saludo cordial.
Give coordinates to the different points: A(u-6,v), B(u,v), C(u,v-13) and D(u-6,v-18). Since B, C and D are on a circle with radius r you get 3 equations with 3 unknowns u,v,r. The equations are: u^2+v^2=r^2; u^2+(v-13)^2=r^2; (u-6)^2+(v-18)^2=r^2. Out of the first 2 equations you get v= 13/2 and using the third equation you get u=21/2. Using then the first equation you get r = sqrt(610)/2.
18-13=5 18*5=x*6 6x=90 x=15
(15+6)/2=21/2 (18+5)/2=23/2
R²=(21/2-6)²=(23/2)² R²=81/4+529/4=610/4 Radius of the circle = √610/2
I think it’s much simpler to use the formula for a circle. (x-a)^2 + (y-b)^2 = r^2 where a and b are the center coordinates of the circle.
The intersection points are (0,0),(6,5)and(6,18) (arbitrary origin on bottom point).
Then solve the three equations generated by the three intersection points. Easy!
(x-h)^2+(y-k)^2=r^2 B(x,y) = 0,0 C=0-13 D=-6,-18
then h^2+k^2=r^2 ...... eq1
h^2+-(13-k)^2=r^2 ...........eq2
(-6-h)^2+(-18-k)^2=r62 ......eq3
solve eq1=eq2 k=169/26
solve eq3=eq1 360-12h+h^2-36k+k^2=r^2 suitation k h=[360-(36*19/26)]/12
suitation h k eq1 r=12.349 As same answer
By symmetry, AF = 5, and by chords theorem AP= 15 and AB =6 given. Then form ∆ PFB, the radius is the product of diagonal ( PF * FB) divided by the double the height of ∆ (2*5)
Hence,( √(15^2+5^2) * √(6^2+5^2))/(2*5)
√250*√61/10 = 5*√610/10 = √610/2
Let O be the center of the circle. Draw radius OT, so that OT is perpendicular to BC and intersects it at M. As OT is perpendicular to BC and BC is parallel to AD, OT is also perpendicular to AD, and intersects it at N.
As BC is a chord, and as OT is an intersecting radius that is perpendicular to BC, then OT bisects BC, thus BM = MC = 13/2.
As ∠ANM = 90° and ∠NMB = 90°, then ABMN is a rectangle, and thus NM = AB = 6 and AN = BM = 13/2. As AD = 18, ND = 18-13/2 = 23/2.
Extend AD from A to E so that ED is a chord. OT bisects ED for the same reasons as it does BC, therefore EN = ND.
EN = ND
EA + AN = ND
EA + 13/2 = 23/2
EA = 10/2 = 5
By the intersecting chords theorem, if two chords of a circle intersect, then the products of the lengths of each side of the intersection are equal for each chord. Extend AB from A to F, so that FB is a chord. As A is the intersection point between FB and ED, then FA•AB = EA•AD.
FA•AB = EA•AD
6FA = 5(18) = 90
FA = 90/6 = 15
Method 1:
Draw OB, and draw radius OS such that OS is perpendicular to FB and intersects it at P. As above, OS bisects FB, so FP = PB = (15+6)/2 = 21/2. As PB is perpendicular to BC, PB = OM = 21/2. As BM is perpendicular to FB, OP = BM = 13/2.
Triangle ∆OBM:
BM² + OM² = OB²
(13/2)² + (21/2)² = r²
r² = 169/4 + 441/4 = 610/4
r = √(610/4) = √610/2
Method 2:
If two chords intersect perpendicularly, then the sum of the squares of the divided chord segments equals 4 times the radius squared.
AB² + FA² + AD² + EA² = 4r²
6² + 15² + 18² + 5² = 4r²
4r² = 36 + 225 + 324 + 25 = 610
r² = 610/4
r = √(610/4) = √610/2
sin(ADC)=6/sqrt(61); Pitagoras: BD=6*sqrt(10); R=BD/(2*sin(ADC))=6*sqrt(10)/(2*6/sqrt(61))=sqrt(610)/2. That's all.
If I understand these methods, the first method triangulates to find the radius using the Pythagorean Thrm twice then uses a circle theorem and the second method requires applying chords twice. I thnk that this is the first time chords are diffcult to visualize.
On second thought, I understood both methods!!!
Alternative "Insane" Method
For number-crunching lovers only. If you're at all error prone with long complicated equations... don't even bother. If you like spending at least half an hour doing tons of algebra on a single problem... go for it. It took me three attempts, but it does work.
Use chord bisector radius to create two Pythagoras equations.
r² = (r - x)² + (13/2)²
r² = [r - (x + 6)]² + [18 - (13/2)]²
Have fun!
At 13:39 PBC is an inscribed rectangle triangle then CP is a diameter; from 14:58 all you need is to use Pythagoras to find |CP|, and then R=|CP|/2: |CP|=√(21²+13²)=√(610), R= √(610)/2.
Glad I'm not the only one who thought of this. As soon as I saw the diagram, I thought of extending BA and using the right triangle.
Why not just use triangles OFM and OBN? We will have r^2=OM^2+MF^2 and r^2=ON^2+NB^2. NB=13/2=6.5, MF=MD=18-6.5=11.5, ON=OM+6. Let OM=x, then r^2=x^2+11.5^2, r^2=(x+6)^2+6.5^2. Subtracting first from second: 0=(x+6)^2+6.5^2-x^2-11.5^2=2*x*6+6^2+6.5^2-11.5^2;x=((11.5-6.5)(11.5+6.5)-36)/12=(5*18-36)/12=54/12=27/6. Then r=sqrt(27^2/6^2+11.5^2)=sqrt(152.5)=12.349.....
Much faster would be following. If distance from the center to AD we lebel x, than: R^2=x^2 + 11.5^2 and R^2=6.5^2 + (6+x)^2
Two equations and two unknowns.
Why not use Analytic Geometry and intersect 2 cords to get the center of the circle? The 2 lines are defined by perpendicular by-sectors of the 2 cords. It looks these RUclips math teachers hate Analytic Geometry.
I was following your few videos, your solution is good, but you need to improve a bit. That is you first explain the problem, then explain to get the desired result what are the information is needed, and how can get those information. Then you start deriving the solution.
(18]^2= 244 (13)^2 =169 (6)^2 =36 {244A+169B+36C}= 419ABC 419ABC/360°=1.59ABC (ABC ➖ 59ABC+1)
How do you come up with the formula after 15:00
已知三點座標，可以直接求圓面積了。
Second Method until the value of AP=15 is excellent.But then misses the obvious, that Inscribed Angle
R^2=6,5^2+a^2...R^2=(18-6,5)^2+(a-6)^2...a=10,5...R^2=6,5^2+10,5^2=152,5
18, 6, 13 replaced with 17, 7 , 10 for this calculation for get better results.
I use calculator to compute, r = sqrt(9^2 + 23^2) / 2 ≈ 12.35.
after 15:00 :Use Pythagorean theorem on triangle PBC
Yes,
ótimo
nice sir
Excelente solução! 🎉
2nd Method much better thanks
La línea AD pasa por el centro del círculo?
Muito bom e muito obrigado!
I think u did mistake on area of BCD. The height it’s not AB
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EC in triangle DCB is not completely the height
Yes it is.
Please could you explain why?
@@giuseppetortelli The OP explain why they think it isn't, or me explain why it is?
bcd is a obtuse triangle bce is rt angle triangle pl correct me
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The solutions offered lack elegance.
I much rather have my teeth extracted (without anesthetics) than attack this problem.
What was the motivation in offering this problem?
би би
You are wrong area bcd you have taken ab is base which is wrong so the area 39 is wrong
its 39
سخيف