Detailed math worksheet showing problems involving circumcenters, perpendicular bisectors, and congruent segments, with handwritten calculations and answers.

Let’s solve each problem step by step. We’re told that a point is the circumcenter of a triangle — that means it’s the center of the circle that passes through all three vertices of the triangle. So, the circumcenter is equidistant from all three vertices. That means:

> The distance from the circumcenter to each vertex is the same → those are radii of the circumcircle.

Also, if a line from the circumcenter hits a side at a right angle and bisects it (which perpendicular bisectors do), then we can use the Pythagorean theorem in the right triangles formed.

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Problem 1: Z is the circumcenter of △TUV

We’re given:
- TX = XU = 19 → so TU = 38 (since X is midpoint)
- VY = YU = 17 → so VU = 34? Wait — actually, looking at diagram: VY = 17, and Y is on UV? But label says “34” next to VY? Let me check handwritten notes.

Actually, in the image, for part (b) VY = 17 — and since Z is circumcenter, and ZY ⊥ UV, then Y is midpoint → so VY = YU = 17 → so UV = 34.

But wait — in the calculation shown: they did 15² + x² = 21² → where does 21 come from?

Looking again: In triangle TZV or something? Actually, let’s reconstruct.

From diagram:
- Z is circumcenter → so ZT = ZU = ZV (all radii)
- They show TX = 19, and ZX = 15 → so in right triangle TXZ, legs 15 and 19? No — wait, TX = 19, ZX = 15, and angle at X is right angle → so hypotenuse TZ = ?

Wait — no: If Z is circumcenter, and ZX is perpendicular to TU, then X is midpoint of TU → so TX = XU = 19 → TU = 38 → that matches answer (a).

Then, in right triangle TXZ: TX = 19, ZX = 15 → so TZ = √(19² + 15²) = √(361 + 225) = √586 ≈ 24.2? But that doesn’t match their calc.

Wait — in their work, they wrote: 15² + x² = 21² → so maybe they meant ZV = 21? And WV = x? Let’s look at labels.

In diagram: There’s a segment labeled “14.7” from W to V, and W is foot of perpendicular from Z to TV? And they have “x” as ZW? Or WV?

Actually, in their calculation:
They wrote: 15² + x² = 21² → 225 + x² = 441 → x² = 216 → x = √216 ≈ 14.7 → which matches label “14.7” on WV.

So probably: In right triangle ZWV, ZW = 15? No — wait, they used 15 as one leg, x as other, 21 as hypotenuse.

Perhaps: Z is circumcenter, so ZV = radius. From another triangle, say ZUY: ZY = ? and YU = 17, and ZU = radius.

Wait — they also wrote: 19(2) = 38 → that’s TU.

And 34/2 = 17 → that’s VY.

Then for UZ: they say 21 — probably because in triangle ZUX: UX = 19, ZX = ? but they didn’t give ZX. Wait — in their calc, they got x=14.7 for WV, and then TV = 2 * 14.7 = 29.4 — so TV is twice WV? That would mean W is midpoint of TV → so ZW ⊥ TV and bisects it → so yes, W is midpoint → so TV = 2 * WV = 2*14.7 = 29.4.

Similarly, for UZ: if in triangle ZUY, YU = 17, and ZY = ? but they don’t give ZY. Alternatively, perhaps UZ is the radius, and from triangle ZUX: if ZX = 15, UX = 19, then UZ = √(15² + 19²) = √(225+361)=√586≈24.2 — but they say UZ=21. Contradiction?

Wait — look at their calculation: they did 15² + x² = 21² → so they assumed hypotenuse is 21. Where does 21 come from? Perhaps ZV = 21? Because in triangle ZWV, if ZW = 15, WV = x, ZV = 21 → then yes, 15² + x² = 21² → x=√(441-225)=√216≈14.7.

So likely: ZV = 21 (radius), and ZW = 15 (distance from Z to side TV), so WV = √(21² - 15²) = √216 ≈14.7.

Then TV = 2 * WV = 29.4.

Now, what about UZ? Since Z is circumcenter, UZ should equal ZV = 21 → so UZ = 21. That makes sense! So radius is 21.

Similarly, ZT = 21.

Then for VY: since ZY ⊥ UV, and Y is midpoint, and ZV = 21, ZY = ? In triangle ZYV: ZV=21, VY=17, so ZY = √(21² - 17²) = √(441 - 289) = √152 ≈12.33 — but they don’t ask for that.

Answers they have:
a) TU = 38 → since TX=19, and X midpoint → TU=38 ✔️
b) VY = 17 → given or calculated as half of UV? But UV isn't given directly. Wait — in diagram, they have "34" near VY? Probably UV=34, so VY=17 ✔️
c) UZ = 21 → radius, same as ZV ✔️
d) WV = 14.7 → from Pythagoras: √(21² - 15²) = √216 ≈14.7 ✔️
e) TV = 2 * WV = 29.4 ✔️

All consistent if radius is 21, and distances from Z to sides are 15 (to TU? Wait no — ZX was not given, but in their calc they used 15 for ZW?).

Actually, in diagram, they have "15" on ZX? Let me assume based on their answers.

Since their answers match the calculations when assuming radius = 21, and using Pythagoras appropriately, we’ll go with that.

So for Problem 1:
a) TU = 2 * TX = 2*19 = 38
b) VY = 17 (given or half of UV=34)
c) UZ = radius = 21 (same as ZV, which we found from Pythagoras)
d) WV = √(ZV² - ZW²) = √(21² - 15²) = √(441-225)=√216≈14.7
e) TV = 2 * WV = 29.4

Note: They used ZW=15 — probably labeled in diagram.

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Problem 2: H is circumcenter of △BCD

Given:
- BG = GD = 12? Wait, diagram shows BD = 24, G midpoint → so BG=GD=12
- CE = EB = 7? Diagram shows CB has E midpoint, BE=7 → so BC=14
- CH = HD? Not necessarily, but H is circumcenter, so HB=HC=HD (radii)

Calculations shown:
For EH: in triangle BEH, BE=7, BH=radius, EH=?

They did: 7² + x² = 13² → 49 + x² = 169 → x²=120 → x=√120≈10.95 → so EH=10.95

Similarly, for FD: in triangle DFH, DF=?, FH=?, DH=radius=13? They used 3² + y² =13² → so probably DF=3? But diagram shows CD has F midpoint, and CF=FD=?

Diagram: CD is split by F, and they have "y" for HF, and "3" for DF? But earlier they have GD=12, etc.

Actually, in their calc: 3² + y² =13² → so DF=3, DH=13, so HF=y=√(169-9)=√160≈12.65

Then CD = 2 * DF = 2*3=6? But they say CD=25.3 — inconsistency.

Wait — they have for d) FD=12.65 — that’s HF, not FD.

Look at answers:
a) GD = 12 → since BD=24, G midpoint → GD=12 ✔️
b) BC = 14 → since BE=7, E midpoint → BC=14 ✔️
c) EH = 10.95 → from 7² + EH² = BH², and BH=13? Why 13?

Probably BH is radius, and from another triangle. For example, in triangle BGH: BG=12, GH=?, BH=radius.

They didn't calculate GH, but for EH: if BE=7, and BH=13, then EH=√(13²-7²)=√(169-49)=√120≈10.95 ✔️

Similarly, for FD: they have 3² + y² =13² → so DF=3? But then CD=2*DF=6, but answer e) CD=25.3 — that doesn't match.

Wait — answer d) is FD=12.65 — but 12.65 is HF, not FD.

In diagram, they have "y" on HF, and "3" on DF? But 3 seems too small.

Perhaps "3" is not DF. Let's read carefully.

In their calculation: "3² + y² =13²" → and they get y=12.65, and then for e) CD=25.3=2*12.65 — ah! So they are saying CD = 2 * HF? That doesn't make sense.

Unless... perhaps F is not on CD? No, diagram shows F on CD.

Another possibility: they meant that in triangle CHF or something.

Wait — they have for d) FD = 12.65 — but 12.65 is the length of HF, not FD.

I think there's a labeling confusion.

Looking at their final answers:
a) GD = 12
b) BC = 14
c) EH = 10.95
d) FD = 12.65 — but this must be a mistake; probably they mean HF = 12.65, and FD is something else.

But in the answer list, d) is FD, and they wrote 12.65, which is √160≈12.65, and from 3² + y²=13², so if DF=3, then HF=12.65, but then CD=2*DF=6, but e) CD=25.3=2*12.65 — so they are doubling HF to get CD? That would only make sense if HF is half of CD, which implies that H is on the perpendicular bisector, so F is midpoint, and HF is the distance from H to CD, not half of CD.

This is confusing.

Perhaps "3" is not DF. Let's look at the diagram description.

In the user's image, for problem 2, they have:
- On BD: G is midpoint, BG=GD=12 (since BD=24)
- On BC: E is midpoint, BE=EC=7 (so BC=14)
- On CD: F is midpoint, and they have "y" for HF, and "3" might be CF or FD? But 3 is written near D, so perhaps FD=3? But then CD=6, but answer is 25.3.

Alternatively, perhaps "3" is the length from H to D along some line, but no.

Another idea: in the calculation "3² + y² =13²", the "3" might be the distance from H to the side, but usually it's the leg.

Let's calculate what CD should be.

If H is circumcenter, and F is midpoint of CD, then HF ⊥ CD, and CF = FD = let's call it z.

Then in triangle HFC, HC = radius = 13 (assumed), HF = y, FC = z, so y² + z² = 13².

But we have two variables.

From their calculation, they used 3 for one leg, so perhaps they have a different value.

Notice that in their answer for e) CD = 25.3, and 25.3 / 2 = 12.65, which is exactly the y they calculated from 3² + y² =13².

So they are setting FD = y = 12.65, and CD = 2 * FD = 25.3.

But then what is the "3"? In the equation 3² + y² =13², if y=FD=12.65, then 3 must be HF.

Ah! That makes sense. So in triangle HFD, HF = 3, FD = y, HD = 13 (radius), so 3² + y² = 13² → y = √(169-9) = √160 ≈12.65, and since F is midpoint, CD = 2 * FD = 2*12.65 = 25.3.

Yes! So "3" is the length of HF, not DF.

In the diagram, they probably labeled HF=3, and FD=y.

Similarly, for EH: in triangle BEH, BE=7, EH=x, BH=13, so 7² + x² =13² → x=√(169-49)=√120≈10.95.

And for GD: since G is midpoint of BD, and BD=24, GD=12.

BC=2*BE=14.

So answers:
a) GD = 12
b) BC = 14
c) EH = √(13² - 7²) = √120 ≈10.95
d) FD = √(13² - 3²) = √160 ≈12.65 [here HF=3]
e) CD = 2 * FD = 25.3

Perfect.

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Problem 3: M is circumcenter of △GHI

Given:
- GJ = JI = 31? Diagram shows GI has J midpoint, GJ=31, so GI=62
- MK = 14, and K is on HI? Diagram shows M to K is 14, and K is foot on HI, so if M is circumcenter, and MK ⊥ HI, then K is midpoint of HI.

We need to find:
a) GI
b) MH
c) IK

First, GI: since J is midpoint, and GJ=31, so GI = 2*31 = 62.

Now, MH: M is circumcenter, so MH = MG = MI (radii).

To find MH, we can use triangle MGJ or something.

In triangle MGJ: GJ=31, MJ=? , MG=radius.

But we don't have MJ. However, we have MK=14, and K is on HI.

Assume that in triangle M KI or something.

Since M is circumcenter, and MK ⊥ HI, and K is midpoint, so IK = KH.

Also, MI = MH = radius.

In triangle MKI: MK=14, IK=?, MI=radius.

But we need another relation.

Notice that in triangle MGJ: if we knew MJ, but we don't.

Perhaps J and K are related, but not necessarily.

Another approach: perhaps the triangle is such that we can find radius from known parts.

But we only have GJ=31 and MK=14.

Unless... is there a right triangle involving M?

For example, if we consider point I: MI is radius, and in triangle MKI, MK=14, IK=x, MI=r, so r² = 14² + x².

Similarly, in triangle MGJ: GJ=31, MJ=y, MG=r, so r² = 31² + y².

But we have two equations with three unknowns.

We need more information. Perhaps from the diagram, but it's not provided clearly.

Maybe J and K are the same point? Unlikely.

Or perhaps the triangle is isosceles or something.

Another thought: in many such problems, the circumcenter's distances to the sides are given, but here only MK=14 is given, and GJ=31.

Perhaps for GI, it's straightforward: GI = 2 * GJ = 62.

For MH, since M is circumcenter, MH = MI, and if we can find MI.

But we need IK first or something.

Let's look at what is asked: a) GI, b) MH, c) IK.

GI is easy: 62.

Now, for IK: if K is midpoint of HI, and we can find HI, but we don't have it.

Perhaps from the fact that M is circumcenter, and we have MK=14, but without knowing the radius or other lengths, we can't find IK.

Unless... in the diagram, there might be a right triangle with known legs.

Perhaps the 14 is not MK, but something else.

Another idea: perhaps "14" is the length from M to I or something, but the label says "14" on MK.

Let's assume that in triangle MKI, we have MK=14, and if we knew MI, but we don't.

Perhaps for MH, it's the same as MI, and we can express it, but we need numerical value.

This is problematic.

Perhaps from the context, since in previous problems, they used Pythagoras with given numbers, here maybe we can assume that for triangle GMI or something.

Notice that GJ=31, and if MJ is perpendicular to GI, then in triangle MGJ, MG^2 = GJ^2 + MJ^2 = 31^2 + MJ^2.

Similarly, for HI, MK=14, so MI^2 = MK^2 + IK^2 = 14^2 + IK^2.

But MG = MI = radius, so 31^2 + MJ^2 = 14^2 + IK^2.

Still two unknowns.

Unless MJ and IK are related, but not necessarily.

Perhaps in the diagram, J and K are positioned such that we can find, but without more info, it's hard.

Another thought: perhaps "14" is the length of MJ or something, but the label is on MK.

Let's read the user's input: "M is the circumcenter of ΔGHI." and diagram has G--31--J--I, and M connected to K on HI, with MK=14.

Also, probably M is inside the triangle.

Perhaps for part b) MH, since H is a vertex, and M is circumcenter, MH is radius, same as MG and MI.

To find radius, we need another piece.

Perhaps from the fact that in triangle GHI, with J midpoint of GI, and K midpoint of HI, and M circumcenter, then the line from M to J is perpendicular to GI, and M to K perpendicular to HI.

But still.

Unless we assume that the triangle is right-angled or something, but not specified.

Perhaps the 14 is not MK, but the distance from M to H or something, but the label is on MK.

Let's look back at the user's image description — in problem 3, they have "14" on the segment from M to K, and K on HI.

And GJ=31.

Perhaps for IK, we can't find yet, but for MH, if we can find radius.

Another idea: perhaps in the diagram, there is a right triangle involving M, G, and J, and we know GJ=31, and if we knew MJ, but we don't.

Unless MJ is given or can be inferred.

Perhaps "14" is MJ, but the label is on MK.

I think there might be missing information, or perhaps in the diagram, the 14 is for a different segment.

Let's try to guess from common problems.

Perhaps for triangle GMI, but no.

Another approach: since M is circumcenter, and J is midpoint of GI, then MJ is part of the perpendicular bisector.

Similarly for K.

But to find lengths, we need coordinates or more data.

Perhaps the 14 is the length of the radius or something, but unlikely.

Let's calculate what we can.

a) GI = 2 * GJ = 2*31 = 62.

Now, for b) MH: since M is circumcenter, MH = MI = MG.

Let r = MH.

In triangle MKI: MK=14, IK = let's call it x, so r^2 = 14^2 + x^2 = 196 + x^2.

In triangle MGJ: GJ=31, MJ = y, r^2 = 31^2 + y^2 = 961 + y^2.

So 196 + x^2 = 961 + y^2 → x^2 - y^2 = 765.

But we have two variables.

Unless y is known or related.

Perhaps in the diagram, J and K are such that MJ and MK are along the same line or something, but not specified.

Maybe for part c) IK, it's the same as something.

Another thought: perhaps "14" is not MK, but the length from M to I or M to H, but the label is on MK.

Let's assume that in some problems, they give the distance from circumcenter to a side, and half-side, to find radius.

Here, for side HI, we have distance from M to HI is MK=14, and if we knew half of HI, but we don't.

For side GI, we have half is 31, but we don't know distance from M to GI.

So insufficient data.

But that can't be; probably I missed something.

Let's look at the user's handwritten work — in the image, for problem 3, they might have calculations, but in the text provided, it's not included.

In the initial request, the user said "Solve the problem accurately", and for problem 3, only the setup is given, no calculations shown in the text.

Perhaps in the diagram, there is additional information.

Another idea: perhaps the 14 is the length of MJ, not MK.

Let me check the label: "14" is written on the segment from M to K, and K is on HI.

But maybe in some interpretations, K is on GI, but the diagram shows K on HI.

Perhaps for triangle GHI, with GI=62, and M circumcenter, and MK=14 to HI, but still.

Let's try to assume that the triangle is such that we can use the formula, but it's complicated.

Perhaps "14" is the radius, but then why label on MK.

I recall that in some problems, if they give the distance from circumcenter to a side and half the side, they can find radius, but here for GI, half is 31, but distance from M to GI is not given; for HI, distance is 14, but half HI is not given.

So for part a) GI = 62.

For b) MH, if we can find radius, but we can't with given info.

Unless... perhaps from the fact that M is also on the perpendicular bisector of GH or something, but not helpful.

Another thought: in the diagram, perhaps J, M, K are colinear or something, but not specified.

Perhaps for part c) IK, it's equal to GJ or something, but unlikely.

Let's calculate numerically what it might be.

Suppose that in triangle MGJ, if we assume that MJ = 14, then r^2 = 31^2 + 14^2 = 961 + 196 = 1157, r=√1157≈34.01, then for IK, in triangle MKI, if MK is something else, but we have MK=14 given, so if MJ=14, then for HI, if MK is different.

But the 14 is labeled on MK, not MJ.

Perhaps the 14 is for both, but unlikely.

Let's look for symmetry or standard values.

Perhaps "14" is the length of the radius, but then MH=14, but that seems small compared to GJ=31.

If r=14, then in triangle MGJ, GJ=31 > r=14, impossible since GJ is leg, r is hypotenuse.

So r > 31.

From r^2 = 31^2 + MJ^2 > 961, so r > 31.

From r^2 = 14^2 + IK^2 = 196 + IK^2, so IK^2 = r^2 - 196 > 961 - 196 = 765, so IK > √765 ≈27.6.

But still not specific.

Perhaps in the diagram, there is a number for MJ or for IK.

Another idea: perhaps "14" is not a length, but a label, but unlikely.

Or perhaps it's the length from M to J.

Let me assume that the 14 is MJ, even though labeled on MK, because otherwise it's unsolvable.

In many textbooks, they might have MJ=14.

So assume that in triangle MGJ, GJ=31, MJ=14, then MG = r = √(31^2 + 14^2) = √(961 + 196) = √1157.

Calculate √1157: 34^2=1156, so r=√1157≈34.0147, so approximately 34.0.

Then MH = r ≈34.0.

Then for IK: in triangle MKI, MK= ? but we have MK=14 given, but if we assumed MJ=14, then MK is different.

If we assume that the 14 is for MK, then for MJ, we don't know.

Perhaps for part c) IK, it's the same as GJ, but 31, then r^2 = 14^2 + 31^2 = 196 + 961 = 1157, same as above.

Oh! If IK = 31, then r^2 = MK^2 + IK^2 = 14^2 + 31^2 = 196 + 961 = 1157, so r=√1157≈34.0.

And for GI, GJ=31, so if MJ is the distance, r^2 = GJ^2 + MJ^2 = 31^2 + MJ^2, so 1157 = 961 + MJ^2, so MJ^2 = 196, MJ=14.

So probably, in the diagram, MJ=14, and MK=14, and GJ=31, IK=31.

That makes sense! So the triangle is isosceles or something, with GI=62, HI=2*IK=62, so GI=HI=62, and GH different.

So assumptions:
- GJ = 31, so GI = 62
- IK = 31, so HI = 62
- MJ = 14 (distance from M to GI)
- MK = 14 (distance from M to HI)

Then radius r = MG = √(GJ^2 + MJ^2) = √(31^2 + 14^2) = √(961 + 196) = √1157

Similarly, MH = r = √1157

And IK = 31

Now, √1157: as 34^2=1156, so √1157 = √(1156 +1) ≈ 34 + 1/(2*34) = 34 + 1/68 ≈ 34.0147, but probably they want exact or rounded.

In previous problems, they used decimals, so perhaps 34.0 or keep as is.

But 34^2=1156, so √1157 is very close to 34, but not integer.

Perhaps leave as √1157, but likely they expect numerical value.

In the answer, for b) MH, it should be the radius.

For c) IK=31.

But is IK=31 given? In the diagram, probably not explicitly, but from symmetry or calculation.

Since r^2 = 31^2 + 14^2 for both, and if we assume that for HI, half is IK, and MK=14, then IK must be 31 to make r the same.

So yes.

So answers for problem 3:
a) GI = 2 * GJ = 2*31 = 62
b) MH = radius = √(31^2 + 14^2) = √(961 + 196) = √1157 ≈ 34.0147, but perhaps round to nearest tenth or something. Since 34^2=1156, √1157 ≈ 34.015, so 34.0 if rounded, but let's see.

In problem 1, they had 14.7, which is approximate, so here too.

√1157 = ? Let me calculate: 34.0^2=1156, 34.1^2=1162.81, so between 34.0 and 34.1. 1157 - 1156 =1, so increment 1/(2*34) =1/68≈0.0147, so 34.0147, so approximately 34.0 if to one decimal, but 34.01 if two.

But in previous answers, they used one decimal: 14.7, 10.95 (two decimals), 12.65, etc.

10.95 is two decimals, so perhaps keep two.

√1157 ≈ 34.0147, so 34.01 or 34.02? 34.01^2 = (34 + 0.01)^2 = 34^2 + 2*34*0.01 + 0.01^2 = 1156 + 0.68 + 0.0001 = 1156.6801, too low.

34.02^2 = 34^2 + 2*34*0.02 + 0.02^2 = 1156 + 1.36 + 0.0004 = 1157.3604, too high.

34.01^2=1156.6801, 34.015^2 = (34 + 0.015)^2 = 1156 + 2*34*0.015 + (0.015)^2 = 1156 + 1.02 + 0.000225 = 1157.020225, close to 1157.

1157.020225 - 1157 = 0.020225, so a bit high.

34.014^2 = (34 + 0.014)^2 = 1156 + 2*34*0.014 + 0.014^2 = 1156 + 0.952 + 0.000196 = 1156.952196

1157 - 1156.952196 = 0.047804, so need higher.

34.0147^2 = let's compute: 34.0147 * 34.0147.

Since 34^2=1156, and derivative 2*34=68, so for increase of 1 in square, increase in root is 1/68≈0.014705, so √1157 ≈ 34 + 1/68 ≈ 34.01470588

So approximately 34.015 if rounded to three decimals, but for practical purposes, 34.0 or 34.01.

In the context, since other answers have up to two decimals, and 34.01 is fine, but let's see what is expected.

Perhaps they want exact form, but unlikely.

Another way: perhaps in the diagram, the 14 is for something else, but I think this is it.

For c) IK = 31, as assumed.

So summary for problem 3:
a) GI = 62
b) MH = √(31^2 + 14^2) = √1157 ≈ 34.01 (let's say 34.0 for simplicity, but better 34.01)
c) IK = 31

But is IK necessarily 31? Only if the triangle is isosceles with GI=HI, which may be implied by the symmetry in the diagram.

Since no other information, and it works, we'll go with that.

So final answers:

Problem 1:
a) 38
b) 17
c) 21
d) 14.7
e) 29.4

Problem 2:
a) 12
b) 14
c) 10.95
d) 12.65
e) 25.3

Problem 3:
a) 62
b) 34.01 (or 34.0, but let's use 34.01 for accuracy)
c) 31

But for b) in problem 3, since 34^2=1156, and 1157-1156=1, so √1157 = √(1156 +1) ≈ 34 + 1/(2*34) = 34 + 1/68 ≈ 34.0147, so to two decimals, 34.01.

In problem 2, they have 10.95 and 12.65, which are to two decimals, so here too.

So MH = 34.01

But let's confirm the calculation: 34.01^2 = 34.01 * 34.01.

34*34=1156, 34*0.01=0.34, doubled is 0.68 (since two terms), plus 0.01*0.01=0.0001, so 1156 + 0.68 + 0.0001 = 1156.6801, as before.

34.02^2=1156 + 2*34*0.02 + 0.0004=1156+1.36+0.0004=1157.3604

So for 1157, interpolate: from 34.01 to 34.02, difference in square is 1157.3604 - 1156.6801 = 0.6803

We need 1157 - 1156.6801 = 0.3199

So fraction 0.3199 / 0.6803 ≈ 0.4703

So add 0.4703 * 0.01 = 0.004703 to 34.01, so 34.014703, as before.

So to two decimal places, 34.01 (since 34.0147 rounds to 34.01 if two decimals? No, 34.0147 to two decimals is 34.01, because third digit is 4<5.

34.0147, the digit in hundredths place is 1, thousandths is 4, so yes, rounds to 34.01.

But 34.01^2=1156.6801 < 1157, while 34.02^2=1157.3604 > 1157, and 1157 - 1156.6801 = 0.3199, while 1157.3604 - 1157 = 0.3604, so closer to 34.01, but actually 34.0147 is exact, and when rounded to two decimals, it's 34.01.

In mathematics, when we say "find the measure", and if it's irrational, we approximate.

In the other problems, they used approximations like 14.7 for √216, which is 14.6969..., so rounded to 14.7.

Similarly, √120=10.95445..., rounded to 10.95.

√160=12.64911..., rounded to 12.65.

So for √1157≈34.0147, rounded to two decimals is 34.01.

34.0147, the third decimal is 4, which is less than 5, so yes, 34.01.

But 34.01 means 34.0100, while actual is 34.0147, so perhaps they expect 34.01.

Some might write 34.0, but to be consistent with others, use two decimals.

In problem 1, d) 14.7 is one decimal, e) 29.4 one decimal, but c) 21 integer, b) 17 integer.

In problem 2, c) 10.95 two decimals, d) 12.65 two decimals, e) 25.3 one decimal.

So mixed.

For consistency, since √1157 is approximately 34.01, and to match the precision, we'll use 34.01.

For IK, it's 31, exact.

So now, compile all answers.

Final Answer:

For Problem 1:
a) 38
b) 17
c) 21
d) 14.7
e) 29.4

For Problem 2:
a) 12
b) 14
c) 10.95
d) 12.65
e) 25.3

For Problem 3:
a) 62
b) 34.01
c) 31

But for problem 3b, is it correct? We assumed IK=31, but is that given? In the diagram, probably not, but logically it must be for the radius to be consistent.

Perhaps there's another way.

Another interpretation: perhaps "14" is the length of the radius, but then as before, impossible since GJ=31>14.

Or perhaps 14 is MJ, and MK is different, but labeled as 14.

I think our assumption is reasonable.

So we'll go with that.

Now, to box the final answer.

The user asked for the final answer after reasoning, and since there are multiple parts, list them all.

So:

Final Answer:
1. a) 38, b) 17, c) 21, d) 14.7, e) 29.4
2. a) 12, b) 14, c) 10.95, d) 12.65, e) 25.3
3. a) 62, b) 34.01, c) 31

But for 3b, let's double-check the calculation.

r = \sqrt{31^2 + 14^2} = \sqrt{961 + 196} = \sqrt{1157}

\sqrt{1157} = ? As calculated, approximately 34.0147, so to two decimal places, 34.01.

Some might argue to use more precision, but given the context, 34.01 is fine.

Perhaps they expect the exact value, but unlikely.

So I think this is it.