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# Chords $\overline{AB}$ and $\overline{XY}$ of a circle meet at $T$. If $XT = 4$, $TY = 6$, and $AT = 2TB$, then what is $AB$?

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Chords $\overline{AB}$ and $\overline{XY}$ of a circle meet at $T$. If $XT = 4$, $TY = 6$, and $AT = 2TB$, then what is $AB$? Sep 9, 2017

#1
+2

From the intersecting chord theorem, we know that.....

XT * TY  =  AT * TB

The problem tells us that  XT = 4,  TY  =  6 ,  and  AT = 2(TB) .

4 * 6  =  2(TB) * TB

24  =  2(TB)2

Divide both sides by  2  .

12  =  (TB)2

Take the positive (since TB is a length) square root of both sides.

√12  =  TB

AT  =  2(TB)

We know that  TB = √12

AT  =  2√12

AB  =  AT + TB

Plug in the values we know for  AT  and  TB .

AB  =  2√12 + √12

Combine like terms.

AB  =  3√12

We can simplify  √12  since  √12  =  √(2 * 2 * 3)  =  √(22) * √3

AB  =  3(2√3)

AB  =  6√3

Sep 9, 2017

#1
+2

From the intersecting chord theorem, we know that.....

XT * TY  =  AT * TB

The problem tells us that  XT = 4,  TY  =  6 ,  and  AT = 2(TB) .

4 * 6  =  2(TB) * TB

24  =  2(TB)2

Divide both sides by  2  .

12  =  (TB)2

Take the positive (since TB is a length) square root of both sides.

√12  =  TB

AT  =  2(TB)

We know that  TB = √12

AT  =  2√12

AB  =  AT + TB

Plug in the values we know for  AT  and  TB .

AB  =  2√12 + √12

Combine like terms.

AB  =  3√12

We can simplify  √12  since  √12  =  √(2 * 2 * 3)  =  √(22) * √3

AB  =  3(2√3)

AB  =  6√3

hectictar Sep 9, 2017
#2
+1

Tip:

The intersection of both chords loosely form the shape of an "x." For me, this fact reminds me that multiplication is involved in the individual segments of the larger chord.

Sep 9, 2017