Solved |CRACK the CODE Arcs + Chords с 14-1 D 2x+1 X2 26 | Chegg.com - Free Printable

Let’s solve each problem step by step. We’ll go one at a time, using circle geometry rules for arcs and chords.

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Problem A:
We have two chords intersecting inside the circle. The segments are 2x+4 and 14 on one chord, and 28 and x on the other.

Rule: When two chords intersect inside a circle, the products of the segments are equal.

So:
(2x + 4) * 14 = 28 * x

Multiply out:
28x + 56 = 28x

Subtract 28x from both sides:
56 = 0 → Wait, that can’t be right.

Wait — let me check again.

Actually, looking at the diagram (even though I won’t describe it), if one chord is split into 2x+4 and 14, and the other into 28 and x, then:

(2x + 4)(14) = (28)(x)

→ 28x + 56 = 28x

→ 56 = 0? That’s impossible.

That means I misread which segments belong to which chord.

Alternative interpretation: Maybe the full chord lengths are given? No — in standard problems like this, when two chords cross, you multiply the parts of each chord.

But here, perhaps the 28 is the whole chord? Let’s think differently.

Wait — maybe the chord with 2x+4 and 14 is one chord, so total length isn't needed — we just use the segments.

But algebra gives 56=0 — contradiction.

Unless... maybe the 28 is not a segment but the whole chord? Then the other part would be 28 - x? But that’s not labeled.

Looking back — perhaps the diagram shows:

One chord: divided into 2x+4 and 14
Other chord: divided into 28 and x

Then product: (2x+4)*14 = 28*x → same as before.

Still 28x + 56 = 28x → 56=0 — impossible.

This suggests an error in setup.

Wait — maybe the 28 is NOT a segment? Or perhaps it's the other way?

Another possibility: Maybe the chord with 28 is split into two parts: one is x, the other is something else? But only x is labeled.

Perhaps the 28 is the entire chord, and it’s split into x and (28 - x)? But that’s not indicated.

Alternatively — maybe the 14 and 2x+4 are on different chords? Let’s try assigning differently.

Suppose:

Chord 1: segments a and b → a*b
Chord 2: segments c and d → c*d
a*b = c*d

If one chord has segments 2x+4 and x, and the other has 14 and 28? Then:

(2x+4)*x = 14*28
→ 2x² + 4x = 392
→ 2x² + 4x - 392 = 0
→ x² + 2x - 196 = 0

Discriminant: 4 + 784 = 788 → sqrt(788) ≈ 28.07 — not integer. Unlikely for school problem.

Wait — perhaps the 28 is the sum? No.

Let me look at common patterns. In many textbooks, they set up so that numbers work nicely.

Try assuming:

Segments: (2x+4) and x on one chord, 14 and 28 on the other? Same as above.

Or: (2x+4) and 28 on one, 14 and x on other?

Then: (2x+4)*28 = 14*x
→ 56x + 112 = 14x
→ 42x = -112 → negative — no.

Another idea: Perhaps the 28 is the diameter or something? But no indication.

Wait — maybe it’s not intersecting chords? But the diagram likely shows two chords crossing.

Perhaps I made a mistake in reading the labels. Let’s assume the correct pairing is:

One chord: 2x+4 and 14
Other chord: 28 and x — but that led to contradiction.

Unless... the 28 is not a segment but the whole chord, and x is half? Not likely.

Wait — what if the chord with 28 is split into two equal parts? But not marked.

I recall that sometimes in these diagrams, if there’s a perpendicular from center, it bisects the chord.

But in Problem A, no center shown.

Perhaps it’s not intersecting chords — maybe it’s secants from outside? But no external point shown.

Let’s skip and come back.

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Problem B:
Right triangle inside circle, with legs 8 and 6, hypotenuse is diameter? Because angle in semicircle is right angle.

So if it’s inscribed in circle and right-angled, then hypotenuse is diameter.

Legs 6 and 8 → hypotenuse = √(6² + 8²) = √(36+64) = √100 = 10

So diameter = 10, radius = 5

But what is x? Probably the radius or something.

Diagram might show x as radius? Or perhaps another segment.

It says “x” near the radius? Assuming x is radius, then x=5.

But let’s confirm.

In circle, if right triangle inscribed, hypotenuse is diameter. So yes, diameter=10, radius=5.

So x=5.

But wait — there’s also a perpendicular from center to chord? Diagram shows a line from center perpendicular to a chord of length 8, and distance from center to chord is x? And another leg is 6?

Let’s reinterpret.

Often in such diagrams: you have a chord of length 8, distance from center to chord is x, and half-chord is 4, and radius is r, and there’s a right triangle with legs x and 4, hypotenuse r.

Also, there’s another right triangle with legs 6 and 8? Confusing.

Perhaps the 6 and 8 are parts of the radius or something.

Standard problem: If you have a chord of length 8, and the distance from center to chord is x, and the radius is r, then by Pythagoras: r² = x² + 4²

Also, if there’s another measurement — say, from end of chord to a point, but here it shows 6 and 8.

Perhaps the 6 is the distance from center to another point.

Another common setup: Two chords perpendicular, etc.

Assume that the radius is composed of two parts: one is x, the other is 6, so radius = x + 6? And it forms a right triangle with half-chord 4.

So: (x + 6)² = x² + 4² ? No, because the distance from center to chord is x, so the radius is hypotenuse, so r² = x² + 4²

But if r = x + 6, then (x+6)² = x² + 16

Expand: x² + 12x + 36 = x² + 16

Subtract x²: 12x + 36 = 16

12x = -20 → negative — impossible.

Perhaps r = 6, and x is distance, then 6² = x² + 4² → 36 = x² + 16 → x² = 20 → x=2√5 — not nice.

Or r = 8, then 64 = x² + 16 → x²=48 — not nice.

Perhaps the 6 and 8 are the legs of the right triangle formed by radius, distance, and half-chord.

For example, if half-chord is 4, and distance is x, radius is r, and r = 6, then 36 = x² + 16 → x²=20.

Not good.

Another idea: The 8 is the full chord, so half is 4. The 6 is the distance from the end of the chord to the foot of the perpendicular or something.

Perhaps it's a different configuration.

Let’s look at the answer choices later; maybe we can infer.

Skip for now.

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Problem C:
Two arcs: 4x° and 4x°, and a central angle or something.

Probably, the two arcs are equal, and together with another arc make 360°, but not specified.

Perhaps it's an inscribed angle subtending those arcs.

Commonly, if two arcs are given, and an angle is formed, but here no angle labeled.

The diagram might show two radii forming angles, and arcs between them.

If two arcs are both 4x°, and they are adjacent, and the remaining arc is something, but not given.

Perhaps the angle at the center is related.

Another thought: If it's a triangle inscribed, but not clear.

Perhaps the 4x and 4x are measures of arcs, and the angle between the chords is half the difference or sum.

But no angle given.

Maybe the total circle is 360°, and there are three arcs: 4x, 4x, and say y, but y not given.

Perhaps the figure is symmetric, and the third arc is also 4x, so 12x = 360, x=30.

That makes sense. Often in such diagrams, if two arcs are labeled 4x and 4x, and it's symmetric, the third might be equal too.

So 4x + 4x + 4x = 360 → 12x=360 → x=30.

Possible.

If not, but let's assume that for now.

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Problem D:
Square inscribed in circle. Side 12, diagonal is diameter.

Diagonal of square = side * √2 = 12√2

So diameter = 12√2, radius = 6√2

But what is x? Probably the radius or something.

Diagram might show x as radius, so x=6√2.

But usually they want numerical value, or perhaps simplified.

Maybe x is the diagonal? But labeled as x on the radius.

Another possibility: In some diagrams, x is the distance from center to side, but for square, that would be half-side = 6.

But let's see.

If square inscribed, center to vertex is radius, center to side is apothem.

Apothem = side/2 = 6 for square? No.

For square, distance from center to side is half the side only if it's aligned, but actually, for a square, the apothem (distance from center to side) is s/2, where s is side.

Is that correct?

No. For a square, the radius (to vertex) is (s√2)/2 = s/√2

The apothem (to side) is s/2.

Yes, because from center to midpoint of side is half the side length if the square is axis-aligned.

For example, square from (-6,-6) to (6,6), side 12, center at origin, distance to side x=6 is 6, which is s/2=6.

Distance to vertex is √(6^2+6^2)=√72=6√2.

So if x is labeled on the apothem, x=6.

If on radius, x=6√2.