Can you find the length AB? | (Two Methods) | #math #maths #geometry

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[ybodoN]

For a fast solution, let's note that B could be the midpoint of ED.
In this case, EB² = DB² = 9² − 6² = 45 and AB = 45 / 6 = 7.5 units.
Pythagoras theorem and the intersecting chords theorem did it!
But this only relies on a special case 🤔 Thank you PreMath! 🙏

[georgexomeritakis]

The second method can be simplified by connecting point C with midpoint M of side DE and apply the Pythagorean theorem twice on the right triangles with hypotenuse 9 and 6

[dirklutz]

The second method is out of this world!

[DaveKube-cxsn]

Nice, interresting and hard math and geometry problems. Good work.

[solimana-soli]

Applying Stewart theorem you get EB*DB, then with intersecting chord theorem AB=15/2

[LuisdeBritoCamacho]

Here I am with my Resolution Proposal.
1) Observe that Triangle [CDE] is not a Right Triangle. It's an Isosceles Triangle with Sides : 9 - 9 - ED.
2) Triangle [BCE] is also an Isosceles Triangle, with Sides : 9 - 9 - 6.
3) The height of Triangle BCE is equal to : h^2 = 9^2 - 3^2 ; h^2 = 81 - 9 ; h^2 = 72 ; h = sqrt(72) ; h = 6*sqrt(2) ; h ~ 8, 5
4) The Area of Triangle [BCE] is equal to : A = 3 * (6*sqrt(2)) ; A = 18*sqrt(2) ; A ~ 25, 5
5) Draw a Line from Point C to intercept the Middle Point between DE; call it C'
6) The Distance between C and C' is equal to : CC' = 36*sqrt(2) / 9 ; CC' = 4*sqrt(2) ; CC' ~ 5, 7
Now,
7) The Distance Between Point B and Point C' (let's name it X) is equal to :
8) h^2 = 36 - X^2 and h^2 = 81 - (9 - X) ^2
9) 36 - X^2 = 81 - 81 + 18X + X^2 ; 18X = 36 ; X = 36/18 ; X = 2
Now we can conclude that the Line ED = 14 ; EB = 9 ; BD = 5
10) 6 * AB = (9 * 5) ; AB = 45 / 6 ; AB = 7, 5 lin un
11) Answer: The Line AB is equal to 7, 5 Linear Units.

[ramanan]

Let angle ECD be 2t.

Let the midpoint of ED be F. Then CF = 9 * cos t. Hence cos(BCF) = (9 * cos t)/6 = 3/2 * cos t.

Let O be the centre of the circle. Since C, F and O are collinear, angle OCB = angle FCB, and cos (OCB) = cos(FCB) = 3/2 * cos t.

Let r be the radius of the circle, we know 9 = 2*r*cos t. Hence r = 9/(2 * cos t). AC = 2*r*cos(OCB) = 2*9/(2 * cos t) * (3/2) * cos t = 13.5

The AB = AC - BC = 13.5 - 6 = 7.5

[wackojacko]

Yes, @ 0:58 With or Without Trigonometry. 🤔 Without Trigonometry human beings would never have been able to travel to the moon. 🙂

[spiderjump]

Triangle ACE is similar to triangle ECB not triangle BCE

[IOSALive]

PreMath, You're awesome! Let's be friends and play together!

[IOSALive]

PreMath, This is great! Let's be friends and have fun together!