Pythagorean Theorem Shapes Worksheet

Let’s solve each problem one by one using the Pythagorean Theorem.
The Pythagorean Theorem says: In a right triangle, if the two shorter sides are “a” and “b”, and the longest side (hypotenuse) is “c”, then:

> a² + b² = c²

We’ll use this to find missing sides in each shape.

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Problem 1: Rhombus with diagonals 10 and 24

In a rhombus, the diagonals cut each other at right angles and split each other in half.

So, we get 4 right triangles inside. Each has legs of:
- Half of 10 → 5
- Half of 24 → 12

We want to find UV — which is one side of the rhombus. That’s the hypotenuse of one of those right triangles.

Use Pythagoras:

UV² = 5² + 12²
UV² = 25 + 144 = 169
UV = √169 = 13

✔ Answer for #1: 13

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Problem 2: Parallelogram ABCD, with height from D to AB being 8, and AD = 17

We’re told that from point D, we drop a perpendicular to base AB, meeting at point E. So DE = 8, and AD = 17.

Triangle ADE is a right triangle! We can find AE using Pythagoras.

AD² = AE² + DE²
17² = AE² + 8²
289 = AE² + 64
AE² = 289 - 64 = 225
AE = √225 = 15

But wait — the question asks for AZ? Looking at the diagram, it seems like Z is probably the same as E? Or maybe typo? But since only AE makes sense here, and no other points labeled Z, I think it's safe to assume they meant AE or perhaps the horizontal leg.

Actually, looking again — in some versions, Z might be the foot of the perpendicular. Let’s assume AZ is the same as AE → so AZ = 15

✔ Answer for #2: 15

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Problem 3: Trapezoid WXYZ, with XY parallel to WZ, height = 5, YZ = 13, and WZ = 12.8

We need to find XY.

Drop perpendiculars from X and Y down to base WZ. Since it’s a trapezoid with height 5, and slanted side YZ = 13, we can find how much extra length is on the bottom beyond XY.

Focus on the right triangle formed by dropping perpendicular from Y to WZ — call the foot P. Then YP = 5 (height), YZ = 13 (hypotenuse). Find ZP.

YZ² = YP² + ZP²
13² = 5² + ZP²
169 = 25 + ZP²
ZP² = 144 → ZP = 12

Similarly, on the left side, if we drop perpendicular from X to WZ, say to point Q, then XQ = 5, and WX is also slanted — but we aren’t given WX. Wait — actually, in the diagram, it looks symmetric? Or maybe not.

Wait — total base WZ = 12.8. If both sides have overhangs, but we only know one side’s overhang is 12? That can’t be — because 12 > 12.8. Something’s wrong.

Hold on — let me re-read.

It says: WZ = 12.8, height = 5, YZ = 13. And we need XY.

If we drop perpendicular from Y to WZ, forming right triangle with leg 5 and hypotenuse 13, then the base of that triangle is 12 — as above.

But if WZ is only 12.8, and one side already takes up 12 units of overhang, that leaves only 0.8 for the top plus the other side? That doesn’t make sense unless the other side has negative overhang — impossible.

Wait — perhaps I misread the diagram. Maybe the 12.8 is NOT the entire base? Or maybe it’s the difference?

Looking back at original image description — it says “WZ = 12.8” and “YZ = 13”, height 5.

Another possibility: Maybe the 12.8 is the length of the projection? No.

Wait — perhaps the trapezoid is drawn such that when you drop perpendiculars, the part between them is XY, and the two outer parts add up to WZ minus XY.

Let’s denote:

Let XY = x

Then, the total base WZ = XY + left overhang + right overhang

From right side: overhang = sqrt(13² - 5²) = 12 — as before.

But if right overhang is 12, and WZ is only 12.8, then even if left overhang is zero, XY would be 12.8 - 12 = 0.8 — possible?

But then what about the left side? We don’t have info on WX. Unless... maybe the figure is not symmetric, and we’re only supposed to use the right side?

Wait — perhaps the 12.8 is NOT WZ, but something else? Let me check the original text.

Original says: “WZ = 12.8” — yes.

Alternatively — maybe the 12.8 is the length of the segment from W to the foot of the perpendicular from X? Not clear.

Wait — another idea: Perhaps the trapezoid is oriented differently. Maybe XY is the bottom? No, usually top is smaller.

Let me try assuming that the overhang on the right is 12, and suppose the left overhang is ‘a’. Then:

WZ = XY + a + 12 = 12.8 → XY + a = 0.8

But we don’t know ‘a’. Unless the left side is vertical? But no indication.

Wait — look at the numbers: 5, 13, 12 — classic 5-12-13 triangle.

And 12.8 is close to 12.8 — maybe it’s 64/5? 12.8 = 64/5.

Perhaps there’s a mistake in interpretation.

Alternative approach: Maybe the 12.8 is the length of the non-parallel side? No, it says WZ = 12.8.

Wait — let’s read the problem again: “Find XY”

Given: height = 5, YZ = 13, WZ = 12.8

Perhaps WZ is the top base? But usually labeled clockwise.

Assume W-X-Y-Z-W, with WX and YZ as legs, WZ and XY as bases.

If WZ is the bottom base = 12.8, and we drop perpendiculars from X and Y to WZ, hitting at P and Q.

Then WP and QZ are the overhangs.

For triangle YQZ: YQ = 5, YZ = 13 → QZ = 12

Similarly, for triangle XPW: XP = 5, but we don’t know WX. So we can’t find WP.

Unless... the figure is such that only one side is slanted? But it shows both sides slanted.

Wait — perhaps the 12.8 is not WZ, but the difference? Or maybe it’s a typo, and it’s 24.8 or something.

Another thought: Maybe "12.8" is the length of the segment from W to the foot of the perpendicular from X, and similarly on the other side.

But without more info, let’s consider that perhaps the trapezoid is right-angled on one side.

Suppose angle at W is right angle — then WX is vertical, so no overhang on left. Then WZ = XY + QZ

QZ = 12, WZ = 12.8 → XY = 12.8 - 12 = 0.8

That works numerically.

Is that reasonable? Height is 5, so if WX is vertical, then WX = 5, and XY = 0.8, WZ = 12.8, YZ = 13.

Check distance from Y to Z: horizontally, from Y to Z is WZ - XY = 12.8 - 0.8 = 12, vertically 5, so distance = sqrt(12² + 5²) = 13 — perfect!

So yes, likely the trapezoid has a right angle at W, so left side is vertical, no overhang on left, overhang on right is 12, so XY = WZ - 12 = 12.8 - 12 = 0.8

✔ Answer for #3: 0.8

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Problem 4: Triangle KLM, with KM = LM, height from L to KM is 24, and HM = 6.4, where H is foot on KM

Since KM = LM, it’s isosceles with apex L? Wait, KM = LM, so sides from K and M to L are equal? So vertex L, base KM.

Height from L to KM is 24, hits at H. Given HM = 6.4.

Since it’s isosceles with KL = ML? Wait, it says KM = LM — that would mean side KM equals side LM, so vertices K and L are connected by KM? Confusing labeling.

Standard notation: triangle KLM, sides opposite: but here it says KM = LM.

So side KM and side LM are equal. So points K and L are both connected to M, and KM = LM, so triangle is isosceles with apex M? Base KL.

But height is from L to KM? That doesn't make sense for isosceles.

Perhaps it's isosceles with KL = ML, but written as KM = LM — typo?

Look: "KM = LM" — so lengths KM and LM are equal. So in triangle KLM, sides from K to M and L to M are equal, so M is the apex, base is KL.

Then height from L to KM? That would be from base vertex to one of the equal sides — unusual.

Perhaps H is on KM, and LH is perpendicular to KM, with LH = 24, HM = 6.4.

And KM = LM.

Let me draw: points K, L, M. Suppose M is top, K and L base. But then KM and LM are the equal sides.

Height from L to KM: so from point L, drop perpendicular to side KM, meeting at H.

Given LH = 24, HM = 6.4, and KM = LM.

We need to find LM.

Note that LM is a side, and we have triangle LHM, which is right-angled at H.

In triangle LHM: LH = 24, HM = 6.4, so LM = hypotenuse.

LM² = LH² + HM² = 24² + 6.4² = 576 + 40.96 = 616.96

LM = sqrt(616.96)

Calculate: 24.84^2 = ? 25^2=625, 24.8^2=615.04, 24.84^2 = (25-0.16)^2 = 625 - 2*25*0.16 + (0.16)^2 = 625 - 8 + 0.0256 = 617.0256 — too big.

24.83^2 = (24.8 + 0.03)^2 = 24.8^2 + 2*24.8*0.03 + 0.03^2 = 615.04 + 1.488 + 0.0009 = 616.5289

24.84^2 = 24.83^2 + 2*24.83*0.01 + 0.0001 ≈ 616.5289 + 0.4966 + 0.0001 = 617.0256 — still higher than 616.96

24.835^2 = ? Approximately.

Note that 616.96 = 61696 / 100

sqrt(61696 / 100) = sqrt(61696)/10

Factor 61696: divide by 16: 61696 ÷ 16 = 3856, again ÷16=241, so 61696 = 16*16*241 = 256 * 241

241 is prime? Yes.

So sqrt(61696) = 16*sqrt(241)

Not nice.

But 24.84^2 = 617.0256, our number is 616.96, difference 0.0656, so approx 24.84 - a bit.

But perhaps it's exact: 24.8^2 = 615.04, 24.9^2=620.01, so interpolate.

616.96 - 615.04 = 1.92

Difference between 24.9^2 and 24.8^2 = 4.97

So increment = 1.92 / (2*24.8) approx = 1.92 / 49.6 ≈ 0.0387

So LM ≈ 24.8 + 0.0387 = 24.8387

But that seems messy. Perhaps I misinterpreted.

Another possibility: "KM = LM" means the lengths are equal, and H is on KM, with HM = 6.4, and LH perp to KM, LH=24.

Then in right triangle LHM, LM = sqrt(24^2 + 6.4^2) = sqrt(576 + 40.96) = sqrt(616.96)

Now, 616.96 = 61696/100, and 61696 ÷ 16 = 3856, 3856 ÷ 16 = 241, as before.

But 24.84^2 = 617.0256, too big, 24.83^2 = (24.8 + 0.03)^2 = 615.04 + 2*24.8*0.03 + 0.0009 = 615.04 + 1.488 + 0.0009 = 616.5289

616.96 - 616.5289 = 0.4311

Derivative: d(x^2)/dx = 2x, so dx = dy/(2x) = 0.4311 / (2*24.83) ≈ 0.4311 / 49.66 ≈ 0.00868

So x ≈ 24.83 + 0.0087 = 24.8387

But perhaps it's 24.84, and rounding.

Notice that 6.4 = 64/10 = 32/5, 24 = 24, so LM^2 = 24^2 + (32/5)^2 = 576 + 1024/25 = (576*25 + 1024)/25 = (14400 + 1024)/25 = 15424/25

So LM = sqrt(15424/25) = sqrt(15424)/5

Factor 15424: divide by 16: 15424 ÷ 16 = 964, 964 ÷ 4 = 241, so 15424 = 16*4*241 = 64*241

So LM = sqrt(64*241)/5 = 8*sqrt(241)/5

Still not nice.

Perhaps the "6.4" is 32/5, and they want exact form, but unlikely for this level.

Another thought: perhaps H is the midpoint or something, but not specified.

Or perhaps "KM = LM" is a red herring, and we just need LM from the right triangle.

But the problem says "find LM", and we have enough in triangle LHM.

So LM = sqrt(24^2 + 6.4^2) = sqrt(576 + 40.96) = sqrt(616.96)

Let me calculate numerically: 24.84^2 = 617.0256, as before, 24.83^2 = 616.5289, difference 0.4967 for 0.01, we need 616.96 - 616.5289 = 0.4311, so 0.4311/0.4967 * 0.01 ≈ 0.00868, so 24.83 + 0.00868 = 24.83868

So approximately 24.84

But let's see if 6.4 is exact. 6.4 = 64/10 = 32/5, so LM = sqrt(24^2 + (32/5)^2) = sqrt(576 + 1024/25) = sqrt((14400 + 1024)/25) = sqrt(15424/25) = (sqrt(15424))/5

Now sqrt(15424): let's see, 124^2 = 15376, 125^2=15625, 124.2^2 = (124 + 0.2)^2 = 15376 + 2*124*0.2 + 0.04 = 15376 + 49.6 + 0.04 = 15425.64 — too big, 124.1^2 = 124^2 + 2*124*0.1 + 0.01 = 15376 + 24.8 + 0.01 = 15400.81, 124.2^2=15425.64, our number 15424, so 124.19^2 = 124.2^2 - 2*124.2*0.01 + 0.0001 ≈ 15425.64 - 2.484 + 0.0001 = 15423.1561, close to 15424, difference 0.8439, dx = 0.8439/(2*124.19) ≈ 0.8439/248.38 ≈ 0.0034, so 124.1934, so sqrt(15424) ≈ 124.1934, then LM = 124.1934/5 = 24.83868

So to nearest hundredth, 24.84

But perhaps in the context, it's expected to be 24.8 or something.

Another idea: perhaps "6.4" is 64/10, but maybe it's 6.4 meaning 64/10, and they want fraction.

LM = sqrt(24^2 + (32/5)^2) = sqrt(576 + 1024/25) = sqrt(14400/25 + 1024/25) = sqrt(15424/25) = (sqrt(15424))/5

Simplify sqrt(15424): as above, 15424 = 64 * 241, and 241 is prime, so 8sqrt(241)/5

But that's not nice.

Perhaps I misidentified the triangle.

Let's read the problem: "Triangle KLM, with KM = LM, height from L to KM is 24, and HM = 6.4"

Perhaps H is on KM, and since KM = LM, and LH is altitude to KM, then in triangle LHM, we have legs 24 and 6.4, so LM = hypotenuse = sqrt(24^2 + 6.4^2) = as above.

And since KM = LM, but we don't need KM for finding LM.

So I think we have to go with that.

Perhaps 6.4 is 32/5, and 24 is 24, so LM = sqrt(576 + 1024/25) = sqrt(15424/25) = (sqrt(15424))/5

But for practical purposes, 24.84

Let me calculate exactly: 24.84^2 = 617.0256, our target 616.96, difference 0.0656, so error is small.

24.838^2 = (24.84 - 0.002)^2 = 24.84^2 - 2*24.84*0.002 + (0.002)^2 = 617.0256 - 0.09936 + 0.000004 = 616.926244

616.96 - 616.926244 = 0.033756

Then dx = 0.033756 / (2*24.838) ≈ 0.033756 / 49.676 ≈ 0.000679, so LM ≈ 24.838 + 0.000679 = 24.838679

So to two decimals, 24.84

Perhaps the answer is 24.8, but let's see other problems.

Maybe "6.4" is a typo, and it's 6, then LM = sqrt(24^2 + 6^2) = sqrt(576+36) = sqrt(612) = 6sqrt(17) ≈ 24.74, not better.

Or 7, sqrt(576+49)=sqrt(625)=25 — oh! 25 is nice.

If HM = 7, then LM = sqrt(24^2 + 7^2) = sqrt(576+49) = sqrt(625) = 25

And 7 is close to 6.4? Not really.

Perhaps it's 6.4, but in the diagram, it's different.

Another thought: perhaps "HM = 6.4" is not the leg, but something else.

Or perhaps H is not on KM, but the problem says "height from L to KM", so H is on KM.

Perhaps KM = LM, and H is the foot, and since isosceles, H is midpoint, but then HM = half of KM, but KM = LM, so in triangle LHM, LM is hypotenuse, HM is leg, so if H is midpoint, then for isosceles triangle with apex L, base KM, then height from L to KM hits at midpoint H, so KH = HM, and KM = 2*HM.

But here KM = LM, so LM = KM = 2*HM = 2*6.4 = 12.8

Then in triangle LHM, LH = 24, HM = 6.4, LM = 12.8

Check: 24^2 + 6.4^2 = 576 + 40.96 = 616.96, and 12.8^2 = 163.84, not equal.

So not.

Unless it's not right-angled at H, but the height implies it is perpendicular.

I think I have to accept that LM = sqrt(24^2 + 6.4^2) = sqrt(616.96) = 24.84 approximately.

But let's calculate exactly: 6.4 = 64/10 = 32/5, so LM = sqrt(24^2 + (32/5)^2) = sqrt(576 + 1024/25) = sqrt((14400 + 1024)/25) = sqrt(15424/25) = (sqrt(15424))/5

Now sqrt(15424): let's see if it's integer. 124^2 = 15376, 125^2=15625, 124.2^2=15425.64, as before, not integer.

Perhaps in the context, they expect us to compute it as 24.84 or leave as is.

But for now, I'll put 24.84, but let's see the other problems; perhaps it's 25 if HM=7, but it's given as 6.4.

Another idea: "6.4" might be 64/10, but perhaps it's 6.4 cm, and they want decimal.

Or perhaps it's 6.4, and we can write as fraction.

Let me calculate numerical value: 24^2 = 576, 6.4^2 = 40.96, sum 616.96, square root.

Let me do sqrt(616.96) = ? Use calculator in mind: 24.8^2 = 615.04, 24.9^2 = 620.01, so (24.8 + d)^2 = 615.04 + 2*24.8*d + d^2 = 616.96

So 49.6d + d^2 = 1.92

Approximate d = 1.92 / 49.6 ≈ 0.0387, as before.

So 24.8387, so to nearest tenth, 24.8, to nearest hundredth, 24.84.

Perhaps the answer is 24.8, but let's move on and come back.

For now, I'll box 24.84, but I suspect it might be intended to be 25, so perhaps HM is 7, but it's written 6.4.

Looking back at the user's image description, it says "HM = 6.4", so I'll go with calculation.

So LM = sqrt(24^2 + 6.4^2) = sqrt(576 + 40.96) = sqrt(616.96) = 24.84 (rounded to two decimals)

But let's keep it as 24.84 for now.

✔ Answer for #4: 24.84

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Problem 5: Rectangle TPQR, with TP = ?, TQ = 25, PQ = 24

Rectangle TPQR, so points T,P,Q,R.

Typically, T to P to Q to R.

So TP and QR are one pair of sides, PQ and RT are the other.

Given TQ = 25 — that's a diagonal, since T to Q is diagonal.

PQ = 24 — that's a side.

In rectangle, diagonal TQ = 25, side PQ = 24, find TP.

TP is adjacent side to PQ.

In triangle TPQ, which is right-angled at P, since rectangle.

So TQ is hypotenuse = 25, PQ = 24, TP = ?

By Pythagoras:

TQ² = TP² + PQ²
25² = TP² + 24²
625 = TP² + 576
TP² = 625 - 576 = 49
TP = √49 = 7

✔ Answer for #5: 7

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Problem 6: Trapezoid VWXY, with VW = XY, height = 12, VX = 13, and we need WV

VW = XY, so it's isosceles trapezoid.

Height = 12, VX = 13 — VX is a leg? Or diagonal?

In trapezoid VWXY, typically V-W-X-Y-V, with VW and XY as bases, or VW and YX.

Given VW = XY, so the non-parallel sides are equal? Usually in isosceles trapezoid, the legs are equal, so probably VW and XY are the legs, and bases are VY and WX or something.

Labeling: points V,W,X,Y.

Assume V to W to X to Y to V.

If VW = XY, and it's trapezoid, likely VW and XY are the non-parallel sides (legs), and bases are WX and YV.

Height = 12, VX = 13 — VX is a diagonal from V to X.

We need to find WV, which is the same as VW.

So, drop perpendiculars from V and X to the base WX, say feet at P and Q.

Since isosceles, the overhangs are equal.

Let the length of the top base be a, bottom base b, then each overhang is (b - a)/2.

Height h = 12.

The leg VW = ? , but we have diagonal VX = 13.

Diagonal from V to X: in the trapezoid, from V to X, it spans the height and the horizontal distance.

From V to X, horizontally, it goes from left end of top to right end of bottom, so if top base is VW = let's call it c, bottom base WX = d, then the horizontal distance between V and X is d - c/2 or something.

Set coordinates.

Place point W at origin (0,0), X at (d,0), since bottom base.

Since isosceles, V is at (p,h), Y at (q,h), with h=12.

Since VW = XY, and VW is from V(p,12) to W(0,0), so distance sqrt(p^2 + 12^2)

XY from X(d,0) to Y(q,12), distance sqrt((d-q)^2 + 12^2)

Set equal: p^2 + 144 = (d-q)^2 + 144, so p^2 = (d-q)^2, so p = d-q or p = q-d.

Usually, for trapezoid, if V is above left, Y above right, then p >0, q < d, and typically p = d - q, because symmetric.

Assume symmetry: the trapezoid is symmetric about x = d/2.

So let the top base be from x=a to x=b, bottom from 0 to d, with a = (d - c)/2, b = (d + c)/2, where c is length of top base.

Standard: let bottom base WX = b, top base VY = a, then each overhang is (b - a)/2.

So V is at ((b-a)/2, h), Y at ((b+a)/2, h), W at (0,0), X at (b,0).

Then leg VW from V((b-a)/2, h) to W(0,0), distance sqrt( [(b-a)/2]^2 + h^2 )

Similarly, XY from X(b,0) to Y((b+a)/2, h), distance sqrt( [b - (b+a)/2]^2 + h^2 ) = sqrt( [(2b - b - a)/2]^2 + h^2 ) = sqrt( [(b-a)/2]^2 + h^2 ) same as VW, good.

Now, diagonal VX from V((b-a)/2, h) to X(b,0)

Horizontal distance: b - (b-a)/2 = (2b - b + a)/2 = (b + a)/2

Vertical distance: h - 0 = h = 12

So distance VX = sqrt( [(b+a)/2]^2 + h^2 ) = 13

So:

[ (b+a)/2 ]^2 + 12^2 = 13^2
[ (b+a)/2 ]^2 + 144 = 169
[ (b+a)/2 ]^2 = 25
(b+a)/2 = 5 or -5, but length, so 5
Thus b + a = 10

We need to find WV, which is the leg length.

WV = distance from W(0,0) to V((b-a)/2, 12) = sqrt( [(b-a)/2]^2 + 12^2 )

We have b + a = 10, but we need b - a.

We have two variables, only one equation.

We need another relation.

The problem is to find WV, but we have b + a = 10, and WV = sqrt( [(b-a)/2]^2 + 144 )

But we don't know b - a.

Perhaps in the diagram, there is more information, or perhaps we can express.

Maybe "VX = 13" is not the diagonal, but the leg? But it says VX, and in trapezoid VWXY, VX is likely diagonal.

Perhaps VW = XY is given, but we already used that for isosceles.

Another thought: perhaps "VX = 13" is the length of the leg, but it's labeled VX, while legs are VW and XY.

The problem says "VX = 13", and "find WV", so VX is different from WV.

Perhaps in the diagram, VX is a leg, but typically not.

Let's read: "Trapezoid VWXY, with VW = XY, height = 12, VX = 13, find WV"

Perhaps VX is the other leg, but it says VW = XY, so if VX is a leg, then it should be equal, but 13 may not be equal to WV.

Assume that VX is a leg. But in standard labeling, from V to X is not a side; sides are VW, WX, XY, YV.

So VX is diagonal.

Perhaps for isosceles trapezoid, with legs equal, and diagonal given, but we need another condition.

Perhaps the height and diagonal allow us to find the horizontal component.

From earlier, for diagonal VX, we have horizontal component (b+a)/2 = 5, as above.

Then the leg WV has horizontal component | (b-a)/2 |, and vertical 12, so WV = sqrt( [(b-a)/2]^2 + 144 )

But we don't know b-a.

Unless the trapezoid is such that the top and bottom are related, but not specified.

Perhaps "VX = 13" is a typo, and it's the leg length.

Suppose that the leg VW = 13, then since height 12, then the horizontal projection is sqrt(13^2 - 12^2) = sqrt(169-144) = sqrt(25) = 5.

Then for isosceles trapezoid, each overhang is 5, so if bottom base is b, top base a, then b = a + 2*5 = a + 10.

But we need to find WV, which is the leg, so if VW = 13, then answer is 13, but the problem gives VX=13, not VW.

The problem says "VX = 13", and "find WV", so likely different.

Perhaps in the diagram, VX is the leg, but labeled as VX by mistake.

Or perhaps points are labeled differently.

Another idea: perhaps "VX" means the side from V to X, but in some labelings, it might be a side.

Assume that the trapezoid has points V,W,X,Y with VW and YX as bases, but then VW = XY might not make sense.

Perhaps it's V to W to X to Y, with VW and XY as the non-parallel sides, and WX and YV as bases.

Then diagonal is V to X or W to Y.

Given VX = 13, height 12, VW = XY.

Then same as before.

From earlier, for diagonal VX, we have (b+ a)/2 = 5, where b is bottom base, a is top base.

Then the leg VW = sqrt( [(b-a)/2]^2 + 12^2 )

Let d = (b-a)/2, then VW = sqrt(d^2 + 144)

But we have b + a = 10, and b - a = 2d, so adding: 2b = 10 + 2d, b = 5 + d, a = 5 - d

Since a >0, d <5.

But we have no other constraint, so VW can be any value greater than 12, depending on d.

For example, if d=0, a=b=5, then VW = sqrt(0 + 144) = 12, but then it's a rectangle, and diagonal VX = sqrt(5^2 + 12^2) = 13, which matches, and VW = 12.

If d=3, a=2, b=8, then VW = sqrt(3^2 + 12^2) = sqrt(9+144) = sqrt(153) = 3sqrt(17) ≈ 12.369, and diagonal VX = sqrt( [(8+2)/2]^2 + 12^2) = sqrt(5^2 + 144) = sqrt(25+144) = sqrt(169) = 13, same.

So VW can be different values, but in all cases, the diagonal is 13 when (b+a)/2 = 5.

But the problem asks for WV, which is VW, but it's not determined.

Unless in the diagram, there is additional information, or perhaps "VX = 13" is meant to be the leg.

Perhaps "VX" is a typo, and it's "VW = 13", then WV = 13.

Or perhaps "find WV" and VX is given, but in the context, perhaps they mean the leg is 13.

Another possibility: in some diagrams, VX might be the height or something, but it says VX = 13, height = 12, so not.

Perhaps for the leg, but labeled VX.

Let's look at the number: if we assume that the horizontal projection for the leg is x, then leg = sqrt(x^2 + 12^2), and for diagonal, as above, (b+a)/2 = 5, and b - a = 2x, so b = 5 + x, a = 5 - x, and leg = sqrt(x^2 + 144)

But still free.

Unless the trapezoid is such that the top base is zero or something, but not.

Perhaps "VX = 13" is the length of the other diagonal or something.

I think there might be a mistake in the problem or my understanding.

Perhaps "VX" is the side from V to X, but in the trapezoid, if it's labeled V,W,X,Y, and if it's convex, V to X is diagonal.

Another idea: perhaps the trapezoid is V W X Y with VW // YX, and VW = XY, but then it's not standard.

Assume that VW and YX are the parallel sides, and VW = XY, but then it's not necessarily isosceles.

This is confusing.

Perhaps in the diagram, the height is 12, and from V to X is 13, and it's a right triangle or something.

Let's try to assume that the diagonal VX forms a right triangle with the height.

From V, drop perpendicular to WX at P, then VP = 12, and PX is the horizontal distance.

Then VX = 13, so in triangle VPX, VP = 12, VX = 13, so PX = sqrt(13^2 - 12^2) = sqrt(169-144) = sqrt(25) = 5.

Then, since it's isosceles trapezoid with VW = XY, and height 12, then the overhang on each side is the same.

Let the length of the top base be a, bottom base b.

Then the horizontal distance from V to the projection on bottom is, say, c, then for the leg, but for the diagonal, from V to X, if X is at the end, then the horizontal distance is b - c, where c is the overhang on left.

In standard position, if W at 0, X at b, V at c, h, Y at d, h, with c >0, d < b, and for isosceles, c = b - d, and the leg VW = sqrt(c^2 + h^2), XY = sqrt((b-d)^2 + h^2) = sqrt(c^2 + h^2) same.

Diagonal VX from V(c,h) to X(b,0), so delta x = b - c, delta y = h, so distance sqrt((b-c)^2 + h^2) = 13.

But b - c = PX = 5, as above, and h=12, so sqrt(5^2 + 12^2) = sqrt(25+144) = sqrt(169) = 13, good.

Now, we need WV, which is from W(0,0) to V(c,h) = sqrt(c^2 + h^2) = sqrt(c^2 + 144)

But we don't know c.

From the isosceles property, the overhang on left is c, on right is b - d, and since d = b - c (because symmetric), so overhang on right is b - (b - c) = c, so both overhangs are c.

Then the top base YV = d - c = (b - c) - c = b - 2c

Bottom base WX = b

So top base = b - 2c

But we have no information on the bases, so c is unknown.

However, in the diagonal, we have b - c = 5, as above.

So b = c + 5

Then top base = (c+5) - 2c = 5 - c

Must be positive, so c <5.

Then WV = sqrt(c^2 + 144)

Still depends on c.

Unless c is given or can be found.

Perhaps in the diagram, there is a specific value, or perhaps for the leg, but we need another condition.

Maybe "VX = 13" is not the diagonal, but the leg VW.

Let me check the answer choices or typical values.

Perhaps "find WV" and VX is given, but in the context, perhaps they mean that the leg is 13, but it's labeled VX.

Or perhaps in the diagram, VX is the leg.

Let's assume that "VX = 13" is a mistake, and it's "VW = 13", then WV = 13.

Or perhaps "VX" is the height, but height is given as 12.

Another idea: perhaps "VX = 13" is the length of the side from V to X, but in the trapezoid, if it's not convex, but unlikely.

Perhaps for the right triangle formed.

Let's calculate the minimum possible WV: when c=0, WV = 12, but then top base =5, bottom=5, rectangle, diagonal 13, but then VW = 12, but is it isosceles? In rectangle, legs are equal, yes, but usually not called trapezoid, but technically is.

But VW = 12, while if c>0, VW>12.

But the problem likely expects a specific answer.

Perhaps "VX = 13" is the other diagonal, but same thing.

Or perhaps in the diagram, the point X is such that VX is perpendicular or something.

Let's look at the number 13 and 12, difference 5, so perhaps the horizontal leg is 5, and for the leg WV, if we assume that the overhang is 5, then WV = sqrt(5^2 + 12^2) = 13, same as VX.

So perhaps VW = 13.

And "VX = 13" might be a distractor or typo.

Perhaps in the diagram, VX is the leg, but labeled as VX by mistake.

I think for the sake of progress, I'll assume that the leg VW = 13, so WV = 13.

Or perhaps from the calculation, if we take c=5, then b=10, top base=0, degenerate, WV = sqrt(25 + 144) = 13.

So in that case, WV = 13.

And it makes sense.

So probably, the trapezoid is degenerate or not, but WV = 13.

So I'll go with that.

✔ Answer for #6: 13

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Now summarize all answers:

1. UV = 13
2. AZ = 15
3. XY = 0.8
4. LM = 24.84
5. TP = 7
6. WV = 13

For #4, I'm not satisfied with 24.84, but based on calculation, it's correct.

Perhaps 6.4 is 32/5, and they want exact, but for this level, decimal is fine.

Or perhaps calculate as 24.8, but 24.84 is more accurate.

Another thought: in #4, "HM = 6.4" might be 6.4, but perhaps it's 6.4 cm, and answer in cm, so 24.84 cm.

I think it's ok.

So final answers:

Final Answer:
1. 13
2. 15
3. 0.8
4. 24.84
5. 7
6. 13