Solved 11. If mZMPL = 63*, find each measure. K a) MIK = d ... - Free Printable

Let’s solve each problem step by step.

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Problem 12:

We’re told:
- m∠AD = (19x + 49)° → This is likely the measure of arc AD.
- mCD = (5x + 37)° → Measure of arc CD.
- We need to find mCD.

Looking at the diagram (even though we can’t see it, from context), points A, B, C, D are on a circle with chords intersecting at E. But since no angles or relationships are given beyond the expressions, and the question asks for mCD, perhaps arcs AD and CD are adjacent and together form a semicircle? Or maybe they’re vertical angles? Wait — actually, in many such problems, if two chords intersect inside a circle, the vertical angles formed are equal to half the sum of the intercepted arcs. But here, we’re not given an angle — just two arc measures.

Wait — re-reading: “If mAD = (19x + 49)° and mCD = (5x + 37)°, find mCD.”

But without more info, this seems unsolvable unless there’s a relationship between the arcs. Perhaps arcs AD and CD are supplementary? Or maybe they are equal? That doesn’t make sense.

Actually — looking again at the diagram description: it shows a circle with chords AC and BD intersecting at E. So likely, angle AEB or something is given? But it’s not written.

Wait — perhaps the problem meant that angle AED or something equals one of those? No.

Alternatively — maybe “mAD” means the measure of arc AD, and “mCD” means arc CD, and perhaps arcs AD and CD together make up a straight line? Like, if A-D-C are consecutive points and ADC is a semicircle? Then arc AD + arc DC = 180°?

That would make sense. Let’s assume that arcs AD and CD are adjacent and form a semicircle (i.e., points A, D, C lie on a semicircle with D between A and C). Then:

arc AD + arc CD = 180°

So:

(19x + 49) + (5x + 37) = 180
→ 24x + 86 = 180
→ 24x = 94
→ x = 94 / 24 = 47/12 ≈ 3.9167

Then mCD = 5x + 37 = 5*(47/12) + 37 = 235/12 + 37 = 235/12 + 444/12 = 679/12 ≈ 56.58°

But let’s check if that makes sense. Arc AD = 19*(47/12) + 49 = 893/12 + 588/12 = 1481/12 ≈ 123.42°, and 123.42 + 56.58 = 180 — yes.

But the problem says “find mCD”, so answer is approximately 56.58°, but maybe exact fraction? The problem doesn’t specify rounding, but others do. Since others say round to nearest hundredth, maybe here too.

But wait — perhaps I misinterpreted. Maybe “mAD” is the measure of angle at A or something? Unlikely.

Another possibility: In some notations, mAD might mean the measure of angle ABD or something — but without diagram, hard to tell.

Given the context of other problems being about arcs and circles, and the fact that 19x+49 and 5x+37 are both linear, likely they are arc measures adding to 180.

So I’ll go with that.

x = 94/24 = 47/12

mCD = 5*(47/12) + 37 = 235/12 + 444/12 = 679/12 ≈ 56.5833... → 56.58°

But let me double-check calculation:

19x + 49 + 5x + 37 = 24x + 86 = 180 → 24x=94 → x=94÷24=4.7083? Wait no:

94 ÷ 24 = 3.91666...

5 * 3.91666 = 19.5833, plus 37 = 56.5833 — yes.

So mCD ≈ 56.58°

But the problem says “find mCD”, and in the blank it’s written as “mCD = _______”, so probably expects numerical value.

I think 56.58 is correct under assumption.

But let’s hold on — maybe the diagram shows that angle at E is given? Not in text.

Perhaps “mAD” is the measure of arc AD, and “mCD” is arc CD, and they are vertical arcs? Doesn't make sense.

Another thought: in circle geometry, when two chords intersect, the measure of the angle formed is half the sum of the intercepted arcs. But here no angle is given.

Unless — perhaps the problem is incomplete? But since it's assigned, likely my first assumption is correct.

I'll proceed with 56.58 for now.

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Problem 13:

Circle with radius 5 mm, central angle 232°. Find length of bolded arc.

Arc length formula: L = (θ/360) * 2πr

θ = 232°, r = 5 mm

L = (232/360) * 2 * π * 5

First, simplify 232/360. Divide numerator and denominator by 8: 29/45

So L = (29/45) * 10π = (290π)/45 = (58π)/9

Now calculate numerically: π ≈ 3.1416

58 * 3.1416 ≈ 182.2128

Divide by 9: ≈ 20.24587...

Round to nearest hundredth: 20.25 mm

Check: 232/360 = 0.6444..., times 2πr = 2*3.1416*5≈31.416, so 0.6444*31.416≈20.24 — yes, rounds to 20.25? Wait:

0.6444 * 31.416 = let's compute:

0.6444 * 30 = 19.332

0.6444 * 1.416 ≈ 0.912

Total ≈ 20.244 → rounds to 20.24? But earlier I had 20.24587, which is closer to 20.25.

Calculate exactly:

232/360 = 29/45

2πr = 10π

So L = (29/45)*10π = 290π/45 = 58π/9

58 * π = 58 * 3.1415926535 ≈ 182.2123739

Divide by 9: 20.24581932...

To nearest hundredth: look at thousandths digit is 5, so round up → 20.25 mm

Yes.

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Problem 14:

Circle with radius 14.5 ft, central angle 73°. Find arc length.

L = (θ/360) * 2πr = (73/360) * 2 * π * 14.5

First, 2 * 14.5 = 29

So L = (73/360) * 29π

Compute 73/360 ≈ 0.202777...

0.202777 * 29 ≈ 5.880555...

Times π ≈ 5.880555 * 3.1415926535 ≈ ?

First, 5.880555 * 3 = 17.641665

5.880555 * 0.1415926535 ≈ let's approximate: 5.88 * 0.1416 ≈ 0.8326

Better: 5.880555 * 0.14 = 0.8232777

5.880555 * 0.0015926535 ≈ 0.00936

Total ≈ 0.8326

So total L ≈ 17.6417 + 0.8326 = 18.4743

More accurately:

73/360 * 29 * π = (73 * 29 * π) / 360

73 * 29 = 2117

So 2117π / 360

2117 * 3.1415926535 ≈ 2117 * 3.1416 ≈ ?

2000*3.1416=6283.2

117*3.1416≈367.5672

Total ≈ 6650.7672

Divide by 360: 6650.7672 / 360 ≈ 18.474353...

Round to nearest hundredth: 18.47 ft

Check: 73/360 ≈ 0.202777, 2πr = 2*3.1416*14.5≈91.1064, then 0.202777*91.1064≈18.474 — yes, rounds to 18.47.

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Problem 15:

Circle with PS = 28 feet. PS is a diameter? From diagram description: points P, Q, R, S, T on circle, center X. PS is labeled as 28 feet, and it looks like a diameter since it goes through center X.

Also, arc PQ is 125°, arc RS is 97°? Wait, the diagram says: near Q it says 125°, near S it says 97° — probably those are arc measures.

Specifically: arc PQ = 125°, arc RS = 97°? But need to interpret.

From the labels: "125°" is near P and Q, so likely arc PQ = 125°.

"97°" is near R and S, so arc RS = 97°.

PS is diameter, so arc PS = 180°.

Points are P, Q, R, S, T around the circle.

Assuming order is P-Q-R-S-T-P or something.

Since PS is diameter, and center X, then arc PS is semicircle, 180°.

Now, part a) find arc ST.

Part b) find arc RPT — which probably means arc from R to P to T, i.e., the long way or short way? Usually, three letters indicate the path, so R to P to T, which might be the major arc.

First, need to find all arc measures.

Given:

- arc PQ = 125° (assumed from label near P and Q)

- arc RS = 97° (label near R and S)

PS is diameter, so arc P to S along the top or bottom is 180°.

Assume the circle is divided into arcs: P to Q, Q to R, R to S, S to T, T to P.

But we have five points, so five arcs.

Sum of all arcs = 360°.

PS is diameter, so if we go from P to S via Q and R, that should be 180°, or via T, also 180°.

Probably, the arc from P to S passing through Q and R is one semicircle, and through T is the other.

Given arc PQ = 125°, and if P to S via Q,R is 180°, then arc QR + arc RS = 180° - arc PQ = 180 - 125 = 55°.

But we're told arc RS = 97°? That can't be, because 97 > 55.

Contradiction.

Perhaps the 125° is arc PT or something else.

Look back: in the diagram description, it says "125°" near P, and "97°" near S, but not specified which arc.

Perhaps 125° is the measure of arc PT or something.

Another interpretation: sometimes the number is placed near the arc it represents.

In the text: "P Q R S T" and "125°" is below P, "97°" is above S.

Perhaps arc PQ is not 125; maybe arc QT or something.

Note that PS = 28 feet, and it's a straight line through center, so diameter = 28 ft, so radius = 14 ft.

Now, for arc lengths, we need central angles.

The key is to find the central angles for the arcs.

From the diagram, likely the 125° is the measure of arc PQ, and 97° is arc RS, but as above, if PS is diameter, and P-Q-R-S is one side, then arc P to S via Q,R should be 180°, so arc PQ + arc QR + arc RS = 180°.

But if arc PQ = 125°, arc RS = 97°, then 125 + 97 = 222 > 180, impossible.

So perhaps the 125° is not arc PQ.

Maybe 125° is the measure of angle at center for arc PT or something.

Another idea: perhaps "125°" is the measure of arc PR or something.

Let's read the problem again: "Using the circle below, find each arc length. Round to the nearest hundredth."

Diagram has points P,Q,R,S,T on circle, center X, PS is diameter (28 ft), and there are numbers: 125° near P, 97° near S.

Perhaps 125° is the measure of arc PQ, and 97° is arc ST or something.

Maybe the 125° is the central angle for arc PQ, and 97° for arc RS, but then we need to find how they fit.

Perhaps PS is diameter, so arc PS = 180°, and the 125° is arc PQ, so if Q is on the semicircle, then arc QS = 180 - 125 = 55°, but then arc RS = 97° doesn't fit.

Unless R is not on the same semicircle.

Perhaps the points are ordered P, T, S, R, Q or something.

Let's assume the circle is divided, and the 125° is the measure of arc from P to Q going the short way, but since PS is diameter, perhaps Q is on the other side.

This is ambiguous.

Another approach: perhaps the 125° is the measure of angle PXQ or something, but usually it's arc measure.

Let's look at part a) find arc ST, and b) arc RPT.

Arc RPT likely means from R to P to T, so the arc not containing S or something.

Perhaps from the diagram, the 125° is arc PT, and 97° is arc RS.

Let me try that.

Suppose arc PT = 125°, arc RS = 97°.

PS is diameter, so arc PS = 180°.

If P to S via T is arc PT + arc TS = 180°, so if arc PT = 125°, then arc TS = 180 - 125 = 55°.

Similarly, if we go P to S via Q and R, arc PQ + arc QR + arc RS = 180°.

But we don't know arc PQ or QR.

We have arc RS = 97°, so arc PQ + arc QR = 180 - 97 = 83°.

But we have five points, so arcs: P to Q, Q to R, R to S, S to T, T to P.

Sum = 360°.

We have arc PT = 125° — but arc PT could be P to T directly, which might be P-Q-R-S-T or P-T directly, but usually arc PT means the minor arc unless specified.

In the diagram, if PS is diameter, and T is on the circle, arc PT could be part of the semicircle.

Perhaps the 125° is the measure of arc from P to Q, and it's on the lower half, and 97° is arc from R to S on the upper half.

Let's calculate the total.

Assume that the arc from P to S via Q and R is 180°, and via T is 180°.

Suppose arc PQ = a, arc QR = b, arc RS = c, arc ST = d, arc TP = e.

a+b+c+d+e = 360°.

PS diameter, so a+b+c = 180° (if P-Q-R-S is one semicircle), and d+e = 180° (S-T-P).

Given that near P it says 125°, which might be arc TP = e = 125°.

Near S it says 97°, which might be arc RS = c = 97°.

Then from a+b+c = 180°, c=97°, so a+b = 83°.

From d+e = 180°, e=125°, so d = 180 - 125 = 55°.

Then sum a+b+c+d+e = 83 + 97 + 55 + 125 = let's add: 83+97=180, 55+125=180, total 360 — perfect.

So arc ST = d = 55°.

Arc RPT: from R to P to T. So from R to P to T. If we go R to S to T to P, or R to Q to P to T? Typically, three letters mean the arc from first to last passing through the middle, so R to P to T means the arc that includes P, so likely R to Q to P to T, or R to S to T to P.

Since P is between R and T in the name, probably it's the arc from R to T passing through P.

So from R to T via P.

Points: R, then to P, then to T.

From R to P: if we go R to Q to P, that's arc RQ + arc QP = b + a.

Then from P to T: arc PT = e = 125°.

But arc from R to T via P would be arc R to P plus arc P to T.

Arc R to P: if going through Q, it's arc RQ + arc QP = b + a = 83° (since a+b=83).

Then arc P to T = e = 125°.

So total arc R to T via P = 83 + 125 = 208°.

The other way from R to T via S is arc RS + arc ST = c + d = 97 + 55 = 152°, which is less, so the minor arc is 152°, but since it's specified as RPT, it probably means the arc passing through P, which is the major arc, 208°.

In some conventions, three letters indicate the specific path, so RPT means start at R, go to P, then to T, so the arc that contains P between R and T.

In this case, from R to T, the arc that goes through P is indeed R-Q-P-T, which is arc RQ + QP + PT = b + a + e = 83 + 125 = 208°.

Yes.

So for part a) arc ST = d = 55°

Part b) arc RPT = 208°

Now, we need arc lengths, not just degrees.

Radius = PS/2 = 28/2 = 14 feet.

Arc length = (θ/360) * 2πr

For a) arc ST: θ = 55°, r=14

L_ST = (55/360) * 2 * π * 14 = (55/360) * 28π

Simplify: 55/360 = 11/72

So (11/72) * 28π = (11 * 28 π) / 72 = (308 π) / 72

Simplify: divide numerator and denominator by 4: 77π / 18

Numerically: 77 * 3.1415926535 / 18 ≈ 241.902634 / 18 ≈ 13.439035...

Round to nearest hundredth: 13.44 feet

For b) arc RPT: θ = 208°, r=14

L = (208/360) * 2 * π * 14 = (208/360) * 28π

Simplify 208/360: divide by 8: 26/45

So (26/45) * 28π = (26 * 28 π) / 45 = 728π / 45

Numerically: 728 * 3.1415926535 / 45 ≈ 2287.079 / 45 ≈ 50.823977...

Round to nearest hundredth: 50.82 feet

Check: 208/360 ≈ 0.5778, 2πr ≈ 87.9646, 0.5778*87.9646≈50.82 — yes.

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Problem 16:

Merry-go-round radius 18 feet. Passenger travels 38 feet along the edge. Find angle of rotation to nearest degree.

Arc length s = r θ, where θ is in radians.

So s = r θ ⇒ θ = s / r = 38 / 18 = 19/9 ≈ 2.1111 radians.

Convert to degrees: θ_deg = θ_rad * (180/π) = (19/9) * (180/π) = (19 * 180) / (9 π) = (19 * 20) / π = 380 / π

π ≈ 3.1415926535, so 380 / 3.1415926535 ≈ 120.957...

Round to nearest degree: 121°

Check: circumference = 2πr = 2*π*18 ≈ 113.097 feet.

38 feet is 38/113.097 ≈ 0.336 of circumference.

0.336 * 360° ≈ 120.96° — yes, rounds to 121°.

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Now, back to Problem 12. I assumed arcs AD and CD sum to 180°, got mCD ≈ 56.58°.

But let me confirm if that makes sense with the diagram. Since it's a common setup, and calculations check out, I'll go with that.

So summary:

12. mCD = 56.58°

13. 20.25 mm

14. 18.47 ft

15a. 13.44 ft

15b. 50.82 ft

16. 121°

But for 12, the answer might be expected as exact or something, but since others round, probably 56.58.

In problem 12, it says "find mCD", and in the blank, likely numerical.

Perhaps they want the expression evaluated.

Another thought: in some geometries, if two chords intersect, the vertical angles are equal, and each is half the sum of the intercepted arcs. But here no angle is given.

Perhaps angle at E is given implicitly. But not in text.

I think my assumption is reasonable.

So final answers:

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Final Answer:
12. 56.58
13. 20.25
14. 18.47
15a. 13.44
15b. 50.82
16. 121