10 6 Practice B Geometry - Fill and Sign Printable Template Online - Free Printable

Let’s solve each problem one by one. These are all about circles and using properties like:

- Intersecting chords: If two chords intersect inside a circle, then the products of the segments of each chord are equal.
→ That is: if chords AB and CD intersect at point P, then AP × PB = CP × PD.

- Secant segments from an external point: If two secants are drawn from a point outside a circle, then:
→ (whole secant) × (external part) = (other whole secant) × (its external part)

- Tangent and secant from an external point: If a tangent and a secant are drawn from a point outside a circle, then:
→ (tangent)² = (whole secant) × (external part)

We’ll apply these rules to each numbered problem.

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Problem 1:

Chords intersect inside circle. Segments: 12 and x on one chord; 18 and 8 on the other.

So:
12 × x = 18 × 8
12x = 144
x = 144 ÷ 12 = 12

✔ Check: 12×12=144, 18×8=144 → correct.

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Problem 2:

Two chords intersect. Segments: 5 and x on one; 3 and 7 on the other.

So:
5 × x = 3 × 7
5x = 21
x = 21 ÷ 5 = 4.2

✔ Check: 5×4.2=21, 3×7=21 → correct.

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Problem 3:

Two chords intersect. Segments: 16 and x on one; 8 and 12 on the other.

So:
16 × x = 8 × 12
16x = 96
x = 96 ÷ 16 = 6

✔ Check: 16×6=96, 8×12=96 → correct.

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Problem 4:

Diameter is 50, so radius is 25. One chord is perpendicular to diameter? Wait — actually, it looks like a chord intersecting the diameter. But looking at diagram: one segment is 8, other is x, and they’re parts of a chord that intersects the diameter which is split into... wait, no — actually, this is likely two chords intersecting: one is the diameter (length 50), broken into two parts? Actually, re-examining: probably the vertical chord is split into 8 and x, and the horizontal diameter is split into two equal parts? No — wait, in standard problems like this, when a chord is perpendicular to a diameter, it gets bisected — but here it's not marked as perpendicular.

Wait — actually, looking again: it’s two chords intersecting: one chord has segments 8 and x, the other chord (the diameter) has segments... but the diameter is 50 total, and if they intersect at center? Not necessarily.

Actually, I think I misread. Let me look again: Problem 4 shows a circle with a horizontal line labeled 50 (probably the diameter), and a vertical chord crossing it, with top segment 8 and bottom segment x. If the vertical chord is perpendicular to the diameter and passes through the center, then it would be bisected — but 8 ≠ x unless x=8, but that doesn’t use the 50.

Wait — perhaps the 50 is the entire diameter, and the intersection point divides the diameter into two parts? But those aren't labeled. Hmm.

Actually, maybe it’s not intersecting chords — perhaps it’s a right triangle inscribed? Or maybe it’s a chord perpendicular to diameter?

Standard theorem: If a diameter is perpendicular to a chord, it bisects the chord. But here, if the chord is split into 8 and x, and if the diameter is perpendicular to it, then 8 = x? But that seems too simple, and why give 50?

Alternatively, perhaps the 50 is the length of the diameter, and the chord is at some distance, forming two segments. But without more info, maybe it’s intersecting chords where the diameter is one chord, split into two parts — but those parts aren’t given.

Wait — perhaps I made a mistake. Looking back at common textbook problems: sometimes in such diagrams, the number 50 is the full diameter, and the chord is perpendicular to it, and we can use Pythagoras.

Assume: The diameter is 50, so radius 25. A chord is perpendicular to the diameter, and the distance from center to chord is, say, d. Then half the chord length is sqrt(r² - d²). But here, the chord is split into 8 and x — if it’s perpendicular to diameter and the diameter passes through center, then the chord should be bisected, so 8 = x? But that contradicts unless x=8.

But that seems odd. Perhaps the 50 is not the diameter? Wait, in the diagram, it’s written along the horizontal line, which spans the circle — so likely diameter.

Another possibility: the two segments 8 and x are parts of a chord that intersects the diameter, and the diameter is divided into two parts by the intersection point. But those parts aren’t labeled. This is confusing.

Wait — let me check online or recall: in many textbooks, problem 4 like this uses the fact that for a chord perpendicular to diameter, but here it might be that the product rule applies if we consider the diameter as being split.

Actually, I think I see it: perhaps the horizontal line is not the diameter but a chord of length 50? But it goes across the circle, so likely diameter.

Perhaps it’s a different setup. Let me skip and come back.

Wait — another thought: in some problems, if you have a chord intersected by a diameter, and you know one segment of the chord and the distances, but here only 8 and x are given for the vertical chord, and 50 for horizontal.

Perhaps the 50 is the sum of the two segments of the diameter? But not specified.

I think there might be a misinterpretation. Let me look at problem 5 for clue.

Problem 5: two secants from external point. External parts 4 and 5, internal parts x and 3? Wait, labels: from external point, one secant has external part 4, then inside circle x, so whole secant is 4+x. Other secant has external part 5, then inside 3, so whole is 5+3=8.

Rule: (external)(whole) = (external)(whole)

So: 4*(4+x) = 5*8
4(4+x) = 40
16 + 4x = 40
4x = 24
x = 6

Okay, that makes sense.

Back to problem 4: perhaps it's similar, but inside the circle. Two chords intersecting. Suppose the horizontal chord is split into a and b, with a+b=50, and vertical chord split into 8 and x. Then by intersecting chords: 8*x = a*b. But we don't know a and b.

Unless the intersection is at the center, then a=b=25, so 8x = 25*25 = 625, x=625/8=78.125, which is larger than diameter, impossible.

If the vertical chord is diameter, but it's labeled with 8 and x, and horizontal is 50, same issue.

Perhaps the 50 is not a chord length but something else. Another idea: in some diagrams, the number on the diameter is the length, and the chord is perpendicular, and we use Pythagoras with radius.

Assume: Circle with diameter 50, so radius 25. A chord is perpendicular to the diameter at a point that is, say, d from center. Then half the chord length is sqrt(25^2 - d^2). But here, the chord is split into 8 and x, so if the foot of the perpendicular is not the midpoint, then it's not bisected.

In general, for a chord intersected by a diameter at a point, the products are equal only if it's the intersecting chords theorem, which requires both chords to be cut into two segments.

Perhaps in problem 4, the horizontal line is a chord of length 50, and it's intersected by another chord that is split into 8 and x, and the horizontal chord is split into, say, m and n, with m+n=50, and 8x = m*n. But without m and n, we can't solve.

This is problematic. Let me look at the answer choices or typical values. Perhaps I missed that the diameter is 50, and the chord is at a distance, but still.

Another thought: in some problems, if a chord is perpendicular to a diameter, and the diameter is 50, and the distance from center to chord is given, but here it's not.

Perhaps the 8 and x are not both on the chord; maybe 8 is from end to intersection, x from intersection to other end, and the diameter is 50, but the intersection point divides the diameter into two parts that are equal if at center, but not specified.

I recall that in some textbooks, for problem 4, it's assumed that the two chords intersect, and the diameter is one chord, split into two equal parts if at center, but 50/2=25, so 8x = 25*25 = 625, x=78.125, which is unreasonable.

Perhaps the 50 is the length of the other chord's segments combined, but not helpful.

Let's move to problem 6 and come back.

Problem 6:

Two secants from external point. One secant: external part 4, internal part 5, so whole secant = 4+5=9. Other secant: external part 3, internal part x, so whole = 3+x.

Rule: (external1)(whole1) = (external2)(whole2)

So: 4 * 9 = 3 * (3 + x)
36 = 3(3 + x)
36 = 9 + 3x
3x = 27
x = 9

✔ Check: 4*9=36, 3*(3+9)=3*12=36 → correct.

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Problem 7:

Two secants from external point. One secant: external part 10, internal part 18, so whole = 10+18=28. Other secant: external part x, internal part 16, so whole = x+16.

Rule: 10 * 28 = x * (x + 16)
280 = x(x + 16)
x² + 16x - 280 = 0

Solve quadratic: discriminant = 256 + 1120 = 1376? Wait, 16^2=256, 4*280=1120, so 256+1120=1376. sqrt(1376) = ? 37^2=1369, 38^2=1444, so not integer. Did I misread?

Labels: from external point, first secant: the part outside is 10, then inside is 18, so yes, whole is 28. Second secant: outside is x, inside is 16, so whole is x+16.

But 10*28=280, x(x+16)=280.

x² +16x -280=0

Divide by 8? Or factor: find two numbers multiply to -280, add to 16. Possible pairs: 20 and -14: 20*(-14)= -280, 20-14=6 no. 28 and -10: 28-10=18 no. 35 and -8: 35-8=27 no. 40 and -7: 40-7=33 no. Not nice.

Perhaps the 18 is the whole secant? Let me check the diagram description. In problem 7, it says "10" and "18" on one secant, "x" and "16" on the other. Typically, the number closer to the external point is the external part.

In many diagrams, for a secant, the external segment is labeled first. So for the first secant, external is 10, then the internal part is 18, so whole is 28. For the second, external is x, internal is 16, whole is x+16.

But 10*28=280, x(x+16)=280.

x² +16x -280=0

Use quadratic formula: x = [-16 ± sqrt(256 + 1120)] / 2 = [-16 ± sqrt(1376)] / 2

sqrt(1376) = sqrt(16*86) = 4sqrt(86), not nice. Probably I have it wrong.

Perhaps the 18 is the whole length of the secant from external point to far end. Let me assume that.

Suppose for the first secant, the whole length is 18, and the external part is 10, then the internal part is 18-10=8. Then for the second secant, whole length is x+16, external is x, internal is 16.

Then rule: external1 * whole1 = external2 * whole2

10 * 18 = x * (x + 16)
180 = x^2 + 16x
x^2 +16x -180=0

Discriminant = 256 + 720 = 976, still not nice.

Another possibility: in some notations, the number on the secant is the length from external point to the first intersection, which is the external part, and the number between the two intersections is the internal part.

In problem 7, it's labeled "10" near the external point, then "18" along the secant inside the circle, so likely external=10, internal=18, whole=28.

Similarly, "x" near external point, "16" inside, so external=x, internal=16, whole=x+16.

But then x(x+16)=280.

Perhaps it's 10 and 18 are both segments, but 18 is not the internal part. Let's calculate numerically.

x^2 +16x -280=0

x = [ -16 ± sqrt(256 + 1120) ] / 2 = [ -16 ± sqrt(1376) ] / 2

sqrt(1376) = sqrt(16*86) = 4sqrt(86) ≈ 4*9.2736 = 37.0944

So x = (-16 + 37.0944)/2 = 21.0944/2 = 10.5472, not nice.

Perhaps the 18 is the distance from external point to the second intersection, so whole secant is 18, external is 10, so internal is 8. Then 10*18 = x*(x+16)? No, the rule is external * whole = external * whole, but for the same point, it's (ext1)*(whole1) = (ext2)*(whole2)

If whole1 = 18, ext1 = 10, then int1 = 8.

For secant2, ext2 = x, int2 = 16, whole2 = x+16.

Then 10 * 18 = x * (x+16) ? No, the rule is ext1 * whole1 = ext2 * whole2, so 10*18 = x*(x+16), same as before.

I think there might be a typo or I need to accept it. But let's look at problem 8.

Problem 8:

Tangent and secant from external point. Tangent length 6, secant has external part 4, internal part x, so whole secant = 4+x.

Rule: (tangent)^2 = (external secant) * (whole secant)

So: 6^2 = 4 * (4 + x)
36 = 4(4 + x)
36 = 16 + 4x
4x = 20
x = 5

✔ Check: 6^2=36, 4*(4+5)=4*9=36 → correct.

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Problem 9:

Two secants from external point. One secant: external part 9, internal part 15, so whole = 9+15=24. Other secant: external part x, internal part 8, so whole = x+8.

Rule: 9 * 24 = x * (x + 8)
216 = x^2 + 8x
x^2 +8x -216=0

Discriminant = 64 + 864 = 928, not nice. sqrt(928)= sqrt(16*58)=4sqrt(58)≈4*7.616=30.464, x= (-8+30.464)/2=11.232, not good.

Perhaps the 15 is the whole secant. Assume external=9, whole=15, then internal=6. Then 9*15 = x*(x+8) ? 135 = x^2 +8x, x^2+8x-135=0, discriminant 64+540=604, not nice.

Another idea: in problem 9, the secant has external part 9, and the part inside is 15, so whole 24, and the other secant has external x, internal 8, whole x+8, so 9*24 = x(x+8) = 216.

x^2 +8x -216=0

Factor: possible roots, try x=12: 144 +96 -216=240-216=24>0, x=10: 100+80-216=180-216=-36<0, x=11: 121+88-216=209-216=-7, x=12: 144+96=240-216=24, so between 11 and 12. Not integer.

Perhaps the 8 is the whole secant. Let's read the diagram: "9" and "15" on one secant, "x" and "8" on the other. Likely 9 is external, 15 is internal for first; x external, 8 internal for second.

But then same as above.

For problem 7, perhaps the 18 is the whole length. Let me assume that for problem 7: external=10, whole=18, so internal=8. Then for second secant, external=x, internal=16, whole=x+16.

Then 10*18 = x*(x+16) ? 180 = x^2 +16x, x^2+16x-180=0.

Discriminant 256+720=976, sqrt(976)=4sqrt(61)≈31.24, x= (-16+31.24)/2=7.62, not good.

Perhaps for problem 7, the 10 and 18 are the two segments of the secant from the external point, but 10 is external, 18 is the rest, so whole is 28, same as before.

I think I need to proceed with what makes sense, and perhaps in the context, for problem 4, it's a special case.

Let's do problem 10.

Problem 10:

Two chords intersect inside circle. Segments: x-1 and x+2 on one chord; 2 and 11 on the other.

So: (x-1)(x+2) = 2*11 = 22

Expand: x^2 +2x -x -2 = 22 → x^2 +x -2 = 22 → x^2 +x -24 = 0

Discriminant = 1 + 96 = 97, not nice. x = [-1±sqrt(97)]/2, not integer.

Perhaps the segments are x-1 and x+2 for one chord, and 2 and 11 for the other, so product equal.

(x-1)(x+2) = 2*11 = 22

x^2 +x -2 = 22, x^2 +x -24=0, as above.

But sqrt(97) is irrational, unlikely for this level.

Perhaps it's (x-1) and (x+2) are the lengths, but maybe x-1 is one segment, x+2 is the other, so their product equals 2*11=22.

Same thing.

Another possibility: in some diagrams, the expressions are for the segments, but perhaps x-1 and x+2 are not both on the same chord; but the problem says "find x", and it's labeled on the chord.

Let's look at problem 11.

Problem 11:

Two chords intersect. Segments: x-1 and x+1 on one chord; 4 and 12 on the other.

So: (x-1)(x+1) = 4*12 = 48

x^2 -1 = 48
x^2 = 49
x = 7 or x = -7, but length can't be negative, so x=7

✔ Check: (7-1)(7+1)=6*8=48, 4*12=48 → correct.

Good.

Problem 12:

Two secants from external point. One secant: external part 10, internal part 20, so whole = 10+20=30. Other secant: external part x, internal part 15, so whole = x+15.

Rule: 10 * 30 = x * (x + 15)
300 = x^2 + 15x
x^2 +15x -300=0

Discriminant = 225 + 1200 = 1425, not nice. sqrt(1425)=5sqrt(57)≈37.75, x= (-15+37.75)/2=11.375, not good.

Perhaps the 20 is the whole secant. Assume external=10, whole=20, then internal=10. Then 10*20 = x*(x+15) ? 200 = x^2 +15x, x^2+15x-200=0, discriminant 225+800=1025, not nice.

Another idea: in problem 12, the secant has external part 10, and the part from external to first intersection is 10, then from first to second intersection is 20, so whole is 30, same as before.

Perhaps for the other secant, x is the external part, 15 is the internal, so whole x+15.

But 10*30=300, x(x+15)=300.

x^2 +15x -300=0

Try x=15: 225 +225 -300=150>0, x=10: 100+150-300= -50<0, x=12: 144+180-300=24>0, x=11: 121+165-300= -14<0, so between 11 and 12.

Not integer.

Perhaps the 15 is the whole secant. Let's assume that for problem 12: external=10, whole=30 for first secant; for second, external=x, whole=15, then internal=15-x, but usually internal is positive, so x<15.

Then 10*30 = x*15
300 = 15x
x = 20, but then internal=15-20= -5, impossible.

So not.

I think there might be a consistent way. Let's go back to problem 4.

Upon second thought, in problem 4, if the horizontal line is the diameter of length 50, and the vertical chord is perpendicular to it, and it is bisected, then 8 = x, but that seems too simple, and why give 50? Unless the 50 is red herring, but unlikely.

Perhaps the 50 is the length of the vertical chord, but it's labeled on the horizontal.

Another common problem: if a chord is perpendicular to a diameter, and the diameter is 50, and the distance from center to chord is d, then half-chord is sqrt(25^2 - d^2). But here, the chord is split into 8 and x, so if the foot is not the center, then the two segments are not equal.

In general, for a chord intersected by a diameter at a point, the products of the segments are equal only if it's the intersecting chords theorem, which requires the diameter to be considered as a chord itself.

So, suppose the diameter is one chord, length 50, so if it's intersected at a point that divides it into a and b, with a+b=50, and the other chord is divided into 8 and x, then 8x = a*b.

But a and b are unknown. However, if the intersection is at the center, a=b=25, then 8x = 625, x=78.125, impossible.

If the intersection is at a distance d from center, then a = 25+d, b=25-d, so a*b = (25+ d)(25- d) = 625 - d^2.

Then 8x = 625 - d^2.

But we have another equation from the chord: the length of the chord is 8+x, and by Pythagoras, (8+x)^2 /4 + d^2 = 25^2, since from center to chord is d, half-chord is (8+x)/2, so [(8+x)/2]^2 + d^2 = 625.

From 8x = 625 - d^2, so d^2 = 625 - 8x.

Plug in: [(8+x)/2]^2 + (625 - 8x) = 625

Simplify: (64 +16x +x^2)/4 + 625 - 8x = 625

Multiply by 4: 64 +16x +x^2 + 2500 - 32x = 2500

So x^2 -16x +64 +2500 -32x = 2500? Let's see:

From:

[(8+x)/2]^2 + 625 - 8x = 625

So [(8+x)/2]^2 - 8x = 0

(64 +16x +x^2)/4 = 8x

Multiply both sides by 4: 64 +16x +x^2 = 32x

x^2 +16x -32x +64 = 0

x^2 -16x +64 = 0

(x-8)^2 = 0

x = 8

Oh! So x=8.

And d^2 = 625 - 8*8 = 625 - 64 = 561, and half-chord = (8+8)/2 = 8, then 8^2 + d^2 = 64 + 561 = 625 = 25^2, yes.

So even though the chord is not bisected in the usual sense, in this case, with the calculation, x=8.

And it makes sense because the equation gave (x-8)^2=0.

So for problem 4, x=8.

Great.

Now for problem 7: let's assume that the 18 is the length from the external point to the second intersection, so whole secant is 18, external is 10, so internal is 8. Then for the other secant, external is x, internal is 16, whole is x+16.

Then rule: external1 * whole1 = external2 * whole2

10 * 18 = x * (x + 16) ? No, the rule is (ext1)*(whole1) = (ext2)*(whole2), so 10*18 = x*(x+16)

180 = x^2 +16x

x^2 +16x -180=0

But earlier calculation showed not nice, but let's solve: discriminant 256 + 720 = 976, not square.

Perhaps for problem 7, the 10 and 18 are the two segments, but 10 is external, 18 is the internal part, so whole is 28, and for the other, x external, 16 internal, whole x+16, so 10*28 = x(x+16) = 280.

x^2 +16x -280=0

Discriminant 256 + 1120 = 1376 = 16*86, so x = [-16 ± 4√86]/2 = -8 ± 2√86, not nice.

Perhaps the 16 is the whole secant. Assume for second secant, whole=16, external=x, then internal=16-x.

Then 10*28 = x*16 ? 280 = 16x, x=17.5, then internal=16-17.5= -1.5, impossible.

Another possibility: in problem 7, the secant has external part 10, and the part between the two intersections is 18, so whole is 28, same as before.

Perhaps the number 18 is labeled on the arc or something, but unlikely.

Let's look at problem 9 again.

Problem 9: secants from external point. One secant: external 9, internal 15, whole 24. Other: external x, internal 8, whole x+8.

9*24 = x(x+8) = 216

x^2 +8x -216=0

Discriminant 64 + 864 = 928 = 16*58, x = [-8 ± 4√58]/2 = -4 ± 2√58, not nice.

Perhaps for problem 9, the 15 is the whole secant. Assume external=9, whole=15, internal=6. Then 9*15 = x*(x+8) ? 135 = x^2 +8x, x^2+8x-135=0, discriminant 64+540=604=4*151, not nice.

Another idea: in some problems, for two secants, if the external parts are a and c, and the internal parts are b and d, then a(a+b) = c(c+d).

For problem 7: a=10, b=18, c=x, d=16, so 10(10+18) = x(x+16) , same as before.

Perhaps the 18 is not the internal part but the distance from the first intersection to the second, which is the internal part, so same.

I think for the sake of time, and since problem 11 worked nicely, perhaps in problem 7, the 18 is the whole length, and 10 is external, so internal=8, and for the other, x external, 16 internal, whole x+16, but then 10*18 = x*(x+16) is not correct because the rule is ext*whole = ext*whole, so it should be 10*18 = x*(x+16) only if x+16 is the whole for the second, which it is, but 180 = x^2 +16x.

But let's calculate the value: x = [ -16 ± sqrt(256 + 720) ] / 2 = [ -16 ± sqrt(976) ] / 2 = [ -16 ± 4√61 ] / 2 = -8 ± 2√61

√61≈7.81, so 2*7.81=15.62, x= -8 +15.62=7.62, not good.

Perhaps for problem 7, the 10 and 18 are on the same secant, but 10 is from external to first intersection, 18 from first to second, so whole 28, and for the other secant, x from external to first, 16 from first to second, so whole x+16, so 10*28 = x*(x+16) = 280.

Then x^2 +16x -280=0

Let me try to factor: possible factors of 280: 10 and 28, 10*28=280, but 28-10=18≠16. 14 and 20, 14*20=280, 20-14=6. 8 and 35, 35-8=27. 7 and 40, 40-7=33. 5 and 56, 56-5=51. 4 and 70, etc. No pair differs by 16.

So perhaps it's 10 and 18 are not both on the secant in that way.

Another thought: in problem 7, the "18" might be the length of the chord inside the circle for the first secant, but the external is 10, so whole 28, same.

Perhaps the number 18 is the distance from the external point to the center or something, but unlikely.

Let's skip to problem 13, which is word problem.

Problem 13:

Key is 10 inches long, inserted 2 inches into hole, so the part inside the ball is 10 - 2 = 8 inches? No.

The key is 10 inches long. It is inserted 2 inches into the hole, so the part from the entrance to the tip is 2 inches inside, but the key extends 8 inches outside? But the problem says: "when inserted 2 inches into the hole, the opposite end of the key passes the ball to the player on the key."

Read carefully: "The T-ballers put the positions of the basketball team members relative to the circular 'key' area marked on the court. If the players' keys pass the ball to the player on the key. What is the length of the goal?"

This is poorly worded. Probably: the key is a rectangle or something, but in basketball, the "key" is the painted area, often rectangular, but here it says "circular 'key'", so perhaps it's a circle.

Then: "if the players' keys pass the ball to the player on the key." Confusing.

Perhaps "keys" means something else. Read: "The T-ballers put the positions... relative to the circular 'key' area... If the players' keys pass the ball to the player on the key."

Perhaps "keys" is a typo, and it's "players".

Another interpretation: there is a circular key area. A player is at a point outside, and throws the ball to a player on the key (on the circumference). The throw is a secant or something.

Then: "What is the length of the goal?" and there is a diagram with a circle, and a line from external point to the circle, with segments.

In the diagram for problem 13, it shows a circle, and from an external point, a line to the circle, with the external part labeled 2 ft, and the part inside the circle labeled 12 ft, and then from the external point, another line to the circle, with external part x, and inside part 4.5 ft, and also a tangent or something.

Looking at the text: "If the players' keys pass the ball to the player on the key. What is the length of the goal?"

And in the diagram, it has: from a point outside, one secant with external part 2 ft, internal part 12 ft, so whole 14 ft. Another secant with external part x, internal part 4.5 ft, so whole x+4.5 ft. And also, there is a tangent from the same external point to the circle, with length 12 ft? In the diagram, it shows a tangent labeled 12 ft, and the secants.

In the user's image description, for problem 13, it says: "13. The T-ballers... " and then "What is the length of the goal?" and in the diagram, there is a circle, and from an external point, a tangent of length 12 ft, and two secants: one with external 2 ft, internal 12 ft; another with external x, internal 4.5 ft.

But the question is "what is the length of the goal?", and x is to be found, but in the diagram, x is labeled on the secant.

Perhaps "the goal" refers to the length x or something.

But in the text, it says "find x" for exercises 1-12, and 13 is separate.

For problem 13, it's a word problem, and we need to find the length of the goal, which might be the diameter or something.

From the diagram: external point P. Tangent PT = 12 ft. Secant PAB, with PA = 2 ft (external), AB = 12 ft (internal), so PB = 14 ft. Another secant PCD, with PC = x (external), CD = 4.5 ft (internal), so PD = x+4.5 ft.

By tangent-secant theorem: (PT)^2 = PA * PB = PC * PD

So 12^2 = 2 * 14 = x * (x + 4.5)

144 = 28? 2*14=28, but 144 ≠ 28, contradiction.

12^2 = 144, PA*PB = 2*14=28, not equal, so mistake.

Perhaps PA is the external part, PB is the whole, so PA*PB = 2*14=28, but 144 ≠ 28.

Unless the 12 ft is not the tangent. In the diagram, it might be that the 12 ft is the internal part or something.

Perhaps for the first secant, the external part is 2 ft, and the whole secant is 12 ft, so internal part is 10 ft. Then PA*PB = 2*12 = 24.

Then for tangent, if it's 12 ft, 12^2=144 ≠ 24.

Perhaps the 12 ft is the length of the tangent, and for the secant, external is 2, internal is 12, so whole 14, then 2*14=28, and 12^2=144, not equal.

Unless the tangent is not from the same point, but it should be.

Another possibility: in the diagram, the 12 ft is labeled on the secant as the internal part, but for the tangent, it's different.

Let's read the user's input: "13. The T-ballers put the positions of the basketball team members relative to the circular 'key' area marked on the court. If the players' keys pass the ball to the player on the key. What is the length of the goal?"

And in the diagram, it shows a circle, and from an external point, a line to the circle with segments 2 ft and 12 ft, and another line with x and 4.5 ft, and also a tangent of 12 ft? But in the text, it says "12 ft" on the tangent, and "2 ft" and "12 ft" on one secant, "x" and "4.5 ft" on the other.

But 2* (2+12) = 2*14=28, and 12^2=144, not equal, so perhaps the 12 ft on the secant is the whole length.

Assume for the first secant, external part is 2 ft, whole secant is 12 ft, so internal part is 10 ft. Then PA*PB = 2*12 = 24.

For the tangent, if it's 12 ft, 144 ≠ 24.

Perhaps the 12 ft is the length of the tangent, and for the secant, the external part is 2 ft, and the internal part is y, so whole 2+y, and 2*(2+y) = 12^2 = 144, so 2+ y = 72, y=70, but in the diagram it's labeled 12 ft for the internal part, not 70.

In the user's description, for problem 13, it says: "12 ft" on the tangent, "2 ft" and "12 ft" on one secant, "x" and "4.5 ft" on the other.

Perhaps the "12 ft" on the secant is the distance from the external point to the second intersection, so whole secant is 12 ft, external is 2 ft, so internal is 10 ft. Then PA*PB = 2*12 = 24.

Then for the tangent, 12^2=144, not 24.

Unless the tangent is not 12 ft; in the diagram, it might be that the 12 ft is on the secant, and the tangent is different.

Perhaps "12 ft" is the length of the goal, and we need to find it, but the diagram has x to find.

I think for problem 13, from the diagram, we have two secants from the same external point, so we can use the secant-secant theorem.

So for problem 13: secant 1: external 2 ft, internal 12 ft, so whole 14 ft. Secant 2: external x ft, internal 4.5 ft, so whole x+4.5 ft.

Then 2 * 14 = x * (x + 4.5)
28 = x^2 + 4.5x
x^2 +4.5x -28 = 0

Multiply by 2 to eliminate decimal: 2x^2 +9x -56 = 0

Discriminant = 81 + 448 = 529 = 23^2

So x = [-9 ± 23] / 4

x = (14)/4 = 3.5 or x = (-32)/4 = -8 (discard)

So x = 3.5 ft

Then the length of the goal might be x or something, but the question is "what is the length of the goal?", and in the context, perhaps the goal is the diameter or the width, but in the diagram, there is also a tangent of 12 ft, but we didn't use it.

In this calculation, we used only the two secants, and got x=3.5, and the tangent is not needed, or perhaps it's for verification.

With x=3.5, then for the tangent, if it exists, (tangent)^2 = 2*14 = 28, so tangent = sqrt(28) = 2sqrt(7) ≈5.29, but in the diagram it's labeled 12 ft, so inconsistency.

Perhaps the 12 ft is not the tangent; in the user's input, for problem 13, it says: "12 ft" on the tangent, but perhaps it's a different label.

To resolve, perhaps in problem 13, the "12 ft" on the secant is the whole length, so external 2 ft, whole 12 ft, internal 10 ft, then 2*12 = 24 = x*(x+4.5)

x^2 +4.5x -24 = 0

2x^2 +9x -48 = 0

Discriminant 81 + 384 = 465, not square.

Or if the 12 ft is the internal part, external 2 ft, whole 14 ft, as before.

Perhaps for the tangent, it's 12 ft, and for the first secant, external 2 ft, internal y, so 2*(2+y) = 144, so 2+y = 72, y=70, but in diagram it's 12 ft, so not.

I think for the sake of completing, and since in many sources, for such problems, we use the secant-secant for problem 13 with the given numbers.

So for problem 13: 2 * (2+12) = x * (x + 4.5) => 2*14 = x(x+4.5) => 28 = x^2 +4.5x

As above, x=3.5

Then the length of the goal might be the diameter or something, but the question is "what is the length of the goal?", and in the context, perhaps it's x, or the 12 ft, but 12 ft is given.

Perhaps "the goal" refers to the length of the secant or something.

Another idea: in basketball, the "goal" might mean the basket, but here it's a circle, so perhaps the diameter.

From the tangent, if we had it, but we don't.

Perhaps in the diagram, the 12 ft is the radius or something.

Let's assume that for problem 13, we are to find x, and the "length of the goal" is x, so 3.5 ft.

But let's box the answers for 1-12 first, and for 13 later.

From earlier:

1. x=12

2. x=4.2 or 21/5

3. x=6

4. x=8 (from our calculation)

5. x=6

6. x=9

7. ?

8. x