Geometry worksheet with problems on triangle midsegments, perpendicular bisectors, and angle bisectors, including diagrams and algebraic expressions.

Let’s solve each problem one by one, step by step.

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Problem 1: Triangle Midsegments

We’re told D, E, F are midpoints of the sides of triangle ABC. That means DE, EF, and FD are midsegments.

A key fact: A midsegment connects two midpoints and is parallel to the third side and half its length.

Looking at the diagram:

- DE = 7 → this is half of BC → so BC = 14
- EF = 12 → this is half of AC → so AC = 24
- DF = 16 → this is half of AB → so AB = 32

Perimeter of triangle ABC = AB + BC + AC = 32 + 14 + 24 = 70

✔ Check: 32+14=46; 46+24=70 → correct.

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Problem 2: Find MN

In triangle JKL, points M and N are midpoints (shown by tick marks on JM and MK, and KN and NL).

So MN is a midsegment → it’s parallel to JL and half its length.

Given:
- MN = 2x + 5
- JL = 8x - 18

Since MN = ½ × JL:

→ 2x + 5 = ½(8x - 18)

Multiply both sides by 2 to eliminate fraction:

→ 2*(2x + 5) = 8x - 18
→ 4x + 10 = 8x - 18

Subtract 4x from both sides:

→ 10 = 4x - 18

Add 18 to both sides:

→ 28 = 4x

Divide by 4:

→ x = 7

Now plug back in to find MN:

MN = 2x + 5 = 2*7 + 5 = 14 + 5 = 19

✔ Check: JL = 8*7 - 18 = 56 - 18 = 38 → half of 38 is 19 → matches.

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Problem 3: Find m∠PST

Triangle PQR has midpoints S and T (tick marks show PS = SQ and QT = TR). So ST is a midsegment → parallel to PR.

That means ∠PST and ∠QPR are corresponding angles → they are equal!

Wait — actually, look again: angle PST is labeled as (11x - 3)°, and angle QPR (at point P) is (5x - 9)°.

But since ST || PR, then ∠PST and ∠QPR are corresponding angles, so they should be equal.

Set them equal:

→ 11x - 3 = 5x - 9

Subtract 5x from both sides:

→ 6x - 3 = -9

Add 3 to both sides:

→ 6x = -6

Divide by 6:

→ x = -1

Wait — negative? Let’s check if that makes sense.

Plug x = -1 into angles:

∠PST = 11*(-1) - 3 = -11 - 3 = -14° → impossible! Angles can’t be negative.

Hmm… maybe I got the relationship wrong.

Actually, looking at the diagram: point S is on PQ, T is on QR. So ST connects midpoints of PQ and QR → so ST is parallel to PR.

Then, angle at S (∠PST) and angle at P (∠QPR) are NOT corresponding — let’s think differently.

Actually, since ST || PR, then line PS is a transversal.

So ∠PST and ∠SPR are alternate interior angles? Wait — no, because ST and PR are parallel, and PS crosses them.

Actually, ∠PST and ∠QPR are on the same side — perhaps they are not directly related that way.

Wait — another idea: Since S and T are midpoints, triangle PST might be similar to triangle PQR? Or maybe we need to use triangle angle sum?

Wait — look at triangle PST. We have two angles given: at P: (5x - 9)°, at S: (11x - 3)°. But we don’t know the third angle.

Alternatively — since ST || PR, then ∠STQ = ∠TRP (corresponding), but that doesn’t help directly.

Wait — here’s the key: In triangle PQR, with midpoints S and T, then ST || PR, so ∠PST = ∠QPR? No — actually, ∠PST and ∠QPR are on different lines.

Let me redraw mentally: Point P, then S on PQ, T on QR. Line ST drawn. Then angle at S inside triangle PST is ∠PST.

Actually, since ST || PR, and PS is part of PQ, then ∠PST and ∠QPR are corresponding angles only if we consider transversal PQ.

Yes! Transversal PQ cuts parallel lines ST and PR. So ∠PST (at S) and ∠QPR (at P) are corresponding angles → they are equal.

But earlier that gave us x = -1, which gives negative angles — impossible.

Unless... maybe the labels are switched? Or maybe I misread the diagram.

Wait — perhaps ∠PST is not the angle at S between P-S-T, but something else? The label says “m∠PST” — that’s angle at S formed by points P, S, T.

Another thought: Maybe triangle PST is isosceles or something? But no info.

Wait — let’s try setting the angles equal anyway and see what happens.

If 11x - 3 = 5x - 9 → x = -1 → angles are negative → invalid.

Perhaps they are supplementary? Because if ST || PR, and PS is transversal, then consecutive interior angles would be supplementary.

Angle at S (∠PST) and angle at P (∠SPR) — but ∠SPR is the same as ∠QPR? Not exactly.

Actually, when two parallel lines are cut by a transversal, consecutive interior angles are supplementary.

Lines ST and PR are parallel. Transversal is PS.

Then the interior angles on the same side are ∠PST and ∠SPR.

But ∠SPR is the angle at P between S, P, R — which is the same as ∠QPR, since S is on PQ.

So yes — ∠PST and ∠QPR are consecutive interior angles → they add up to 180°.

That makes more sense!

So:

(11x - 3) + (5x - 9) = 180

Combine like terms:

16x - 12 = 180

Add 12 to both sides:

16x = 192

Divide by 16:

x = 12

Now find m∠PST = 11x - 3 = 11*12 - 3 = 132 - 3 = 129°

Check other angle: 5x - 9 = 60 - 9 = 51°

129 + 51 = 180 → perfect, supplementary.

✔ Correct.

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Problem 4: Perpendicular Bisector

EF is the perpendicular bisector of CD. That means it cuts CD at its midpoint and forms right angles.

Also, any point on the perpendicular bisector is equidistant from the endpoints of the segment.

Point E is on the perpendicular bisector → so EC = ED.

From diagram:

EC = 4x + 19
ED = 6x - 7

Set equal:

4x + 19 = 6x - 7

Subtract 4x:

19 = 2x - 7

Add 7:

26 = 2x

x = 13

✔ Check: EC = 4*13 + 19 = 52 + 19 = 71; ED = 6*13 - 7 = 78 - 7 = 71 → equal.

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Problem 5: If JK ≅ KL, find JL

JK ≅ KL means triangle JKL is isosceles with JK = KL.

M is the midpoint of JL? Wait — diagram shows M on JL, and KM is drawn. Also, JM = 14x - 9, ML = 8x + 3.

Since JK = KL, and KM is drawn to base JL, then in an isosceles triangle, the median from the apex to the base is also the altitude and angle bisector — but more importantly, it hits the midpoint.

So M is the midpoint of JL → so JM = ML.

Set them equal:

14x - 9 = 8x + 3

Subtract 8x:

6x - 9 = 3

Add 9:

6x = 12

x = 2

Now find JL = JM + ML = (14x - 9) + (8x + 3) = 22x - 6

Plug x=2: 22*2 - 6 = 44 - 6 = 38

Or compute separately: JM = 14*2 - 9 = 28 - 9 = 19; ML = 8*2 + 3 = 16 + 3 = 19; total = 38.

✔ Correct.

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Problem 6: Find QR

Diagram shows quadrilateral PQRS with right angles at P and R, and QS is a diagonal. Also, PS = RS? Wait — tick marks on PS and RS? Actually, looking: there are tick marks on PS and RS? No — wait, the diagram shows:

At P: right angle, side PS = 3x + 22
At R: right angle, side RS = 10x - 41
And QS is common hypotenuse for triangles PQS and RQS? Wait — actually, it looks like triangles PQS and RQS share side QS, and both have right angles at P and R.

Also, there’s a mark at S indicating that angles at S are equal? Or maybe QS is angle bisector?

Wait — actually, the diagram shows that QS is the angle bisector of angle PSR? And also, since angles at P and R are right angles, and QS is common, and if QS bisects angle S, then triangles PQS and RQS are congruent by AAS or ASA?

Actually, standard theorem: If a point is on the angle bisector, and you drop perpendiculars to the sides, then those perpendiculars are equal.

Here, from point Q, we have perpendiculars to SP and SR? Wait — no.

Actually, looking: Points P and R are feet of perpendiculars from Q to lines SP and SR? Not quite.

Better interpretation: Triangles QPS and QRS are both right triangles, sharing hypotenuse QS. Also, angle at S is bisected by QS? The diagram shows arc marks at S, suggesting that angle PSQ equals angle RSQ.

So, in triangles QPS and QRS:

- Right angles at P and R
- Angle at S equal (bisected)
- Side QS common

So by AAS, triangles are congruent → so corresponding legs are equal: PS = RS

Therefore:

3x + 22 = 10x - 41

Subtract 3x:

22 = 7x - 41

Add 41:

63 = 7x

x = 9

Now, QR is a leg of triangle QRS. But we need to find QR.

Wait — the question is "Find QR". But in the diagram, QR is not labeled with expression. However, since triangles are congruent, QR = QP? But we don't have QP.

Wait — perhaps QR is the same as QP? But we need a value.

Actually, re-examining: The problem says "Find QR", and in the diagram, QR is a side, but no expression is given for it. However, since triangles are congruent, and we have PS and RS, but QR is opposite to angle at S in triangle QRS.

Wait — perhaps I misunderstood. Let me read the diagram again.

Actually, in triangle QRS, right-angled at R, with legs QR and RS, hypotenuse QS.

Similarly, in triangle QPS, right-angled at P, legs QP and PS, hypotenuse QS.

Since triangles are congruent, then QR = QP, and RS = PS.

We already used RS = PS to find x=9.

But to find QR, we need more information. Unless... perhaps QR is expressed in terms of x? But it's not labeled.

Wait — looking back at the diagram: Is there a label on QR? In the user's image, under problem 6, it shows:

Left side: PS = 3x + 22
Right side: RS = 10x - 41
And QR is the side from Q to R, which is vertical? But no expression.

Perhaps I made a mistake. Another possibility: Since QS is the angle bisector, and we have right angles, then by angle bisector theorem? But angle bisector theorem applies to triangles, not here.

Wait — actually, in this configuration, since both triangles share hypotenuse and have equal acute angles at S, they are congruent, so all corresponding sides equal.

Thus, QR = QP, but we don't know QP.

But the problem asks for QR — perhaps it's expecting us to realize that QR is not determined? That can't be.

Wait — perhaps "QR" is a typo, and it's supposed to be "RS" or "PS"? But no, it says "Find QR".

Another thought: In the diagram, is QR labeled? Looking at the original image description: for problem 6, it shows "3x+22" on left, "10x-41" on right, and "QR" is the side on the right triangle, but no expression.

Perhaps after finding x, we can find lengths, but QR is not given an expression.

Unless... in triangle QRS, with right angle at R, and RS = 10x - 41, and if we knew QS or something, but we don't.

I think there might be a misinterpretation. Let me search for similar problems.

Standard problem: When you have two right triangles sharing a hypotenuse, and the angle at the vertex is bisected, then the legs adjacent to the bisected angle are equal — which we used: PS = RS.

But to find QR, perhaps it's equal to QP, but still unknown.

Wait — perhaps the diagram implies that QP and QR are the other legs, and since triangles are congruent, QP = QR, but we need a numerical value.

But without additional information, we can't find QR numerically unless it's expressed.

Perhaps "QR" is meant to be the length of the side, and since we have x, and if QR was given an expression, but it's not.

Looking back at the user's input: for problem 6, it says "Find QR." and the diagram has "3x+22" and "10x-41" on the two legs from S, and QR is the side from Q to R.

But in many such problems, once you find x, you can find the lengths, but QR isn't labeled with an expression.

Unless... perhaps QR is the same as the other leg, but no.

Another idea: Perhaps "QR" is a mistake, and it's supposed to be "RS" or "PS", but that doesn't make sense.

Or perhaps in the diagram, QR is labeled, but in the text description, it's omitted.

Given that in problem 7, angles are given, here for problem 6, after finding x=9, we can find PS = 3*9 + 22 = 27 + 22 = 49, RS = 10*9 - 41 = 90 - 41 = 49, so they are equal.

But QR is still unknown.

Perhaps the problem is to find the length of QR, and it's implied that we need to use Pythagoras, but we don't have QS.

I think there might be an error in my reasoning or in the problem interpretation.

Let me try a different approach. Perhaps QS is the angle bisector, and by the angle bisector theorem in triangle PSR or something.

Triangle PSR? Points P, S, R. But Q is outside.

Actually, the figure is kite-shaped or something.

Notice that in the diagram, there are right angles at P and R, and S is the vertex, and Q is connected to S, and also to P and R? The diagram shows lines from Q to P, Q to S, Q to R, and S to P, S to R.

So it's like two right triangles sharing the hypotenuse QS? No, sharing the vertex Q and S.

Actually, it's quadrilateral PQRS with diagonals or something.

Perhaps it's better to assume that since the triangles are congruent, and we have PS = RS = 49 when x=9, and if we assume that QP = QR, but still no value.

Unless the problem is to find the length of the side that is QR, and it's not given, but perhaps in the context, QR is the same as the other leg, but no.

I recall that in some problems, when you have this setup, the length QR can be found if we know more, but here we don't.

Perhaps "Find QR" means find the expression or the value, but since it's not labeled, maybe it's a trick.

Another thought: In the diagram, is QR labeled with an expression? In the user's initial description, for problem 6, it shows "3x+22" on left, "10x-41" on right, and "QR" is written near the right side, but no expression attached to it.

Perhaps QR is the side from Q to R, and in triangle QRS, with right angle at R, and RS = 10x-41, and if we knew angle or something, but we don't.

I think I need to look for a different interpretation.

Let's read the problem again: "6. Find QR." and the diagram has points P, Q, R, S, with right angles at P and R, and QS drawn, and arc at S indicating angle bisector.

In such cases, the distance from Q to the sides are equal, but here the distances are QP and QR, since perpendiculars to the sides.

Yes! That's it.

In angle bisector theorem for a point on the bisector: the perpendicular distances to the two sides are equal.

Here, QS is the angle bisector of angle PSR.

From point Q, we drop perpendiculars to the two sides of the angle. The sides of angle PSR are SP and SR.

The perpendicular from Q to SP is QP (since angle at P is right angle, so QP ⊥ SP).

Similarly, the perpendicular from Q to SR is QR (since angle at R is right angle, so QR ⊥ SR).

Therefore, since Q is on the angle bisector of angle PSR, then the perpendicular distances to the two sides are equal: QP = QR.

But we still don't have their values.

However, in the triangles, we have PS and RS, which are along the sides, not the perpendiculars.

The perpendicular distances are QP and QR, which are the lengths we want, but they are not given expressions.

Unless... perhaps in the diagram, QP and QR are the legs, and we can find them if we had more, but we don't.

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

Perhaps "QR" is meant to be "RS" or "PS", but that doesn't make sense.

Another idea: Perhaps after finding x, we can find the length of QS or something, but the problem asks for QR.

Let's calculate what we can.

With x=9, PS = 3*9 + 22 = 27+22=49, RS = 10*9 - 41 = 90-41=49.

Now, in triangle QPS, right-angled at P, with leg PS = 49, and leg QP = ? , hypotenuse QS.

Similarly for triangle QRS.

But without additional information, we can't find QP or QR.

Unless the diagram implies that QP and QR are equal, which they are, but still no value.

Perhaps the problem is to find the length of the side that is QR, and it's not specified, but in many textbooks, they might expect us to realize that QR = QP, and perhaps give the value if it was labeled, but it's not.

I recall that in some problems, they ask for the length of the bisector or something, but here it's QR.

Perhaps "QR" is a typo, and it's supposed to be "RS" or "PS", but that would be 49, and we have it.

Or perhaps "find the length of QR" and it's the same as QP, but we need to express it.

I think I need to assume that the problem intends for us to find the value of the legs or something else.

Let's look at problem 7 for clue, but it's separate.

Another thought: In the diagram, is there a label on QR? In the user's image, for problem 6, it might be that QR is not labeled, but perhaps in the context, after finding x, we can find it if we assume something.

Perhaps the side QR is the one with expression, but in the text, it's not given.

I think there might be an error, but let's proceed with what we have.

Perhaps "Find QR" means find the length, and since the triangles are congruent, and we have PS = RS = 49, and if we assume that the other legs are equal, but still.

Let's calculate the length if we had more, but we don't.

Perhaps the problem is to find the value of x first, but it says "Find QR".

I recall that in some versions of this problem, they give expressions for QP and QR, but here they don't.

Perhaps in the diagram, QR is labeled with an expression, but in the text description, it's omitted.

Given that, and since we have x=9, and if we assume that QR is to be found, but it's not possible, perhaps the answer is 49, but that's PS or RS.

Let's check online or standard problems.

Upon second thought, in this configuration, the length QR is not determined solely by x; we need more information. But that can't be for a homework problem.

Perhaps "QR" is the side from Q to R, and in the diagram, it's the same as the other leg, but no.

Another idea: Perhaps the arc at S indicates that angle PSQ = angle RSQ, and with right angles, then by trigonometry, but still.

Let's use the fact that in triangle QPS and QRS, they are congruent, so QP = QR, and PS = RS = 49, and if we let QP = QR = y, then by Pythagoras, QS^2 = QP^2 + PS^2 = y^2 + 49^2, and similarly for the other, same thing.

But we have one equation with two unknowns.

So insufficient.

Unless the problem is to find PS or RS, but it says QR.

Perhaps "QR" is a mistake, and it's "RS" or "PS", which is 49.

Or perhaps "find the length of the side that is QR", and it's 49, but that's not accurate.

Let's look at the answer choices or something, but there are none.

Perhaps in the diagram, QR is labeled with an expression involving x, but in the text, it's not mentioned.

Given the constraints, and since we have x=9, and PS=RS=49, and if we must give an answer, perhaps QR is 49, but that's incorrect because QR is the other leg.

Unless in the diagram, the side labeled "10x-41" is QR, but in the text, it's described as on the right, and QR is on the right, so perhaps "10x-41" is the length of QR.

Let's read the user's description: "6. Find QR. 3x + 22 10x - 41" and the diagram has "3x+22" on left, "10x-41" on right, and "QR" is likely the side on the right, so perhaps "10x-41" is the length of QR.

That makes sense! In many diagrams, the expression is written next to the side it represents.

So probably, "10x - 41" is the length of QR.

But then why do we need to find it? We have it in terms of x, but we need numerical value.

And we have x=9 from earlier, so QR = 10*9 - 41 = 90 - 41 = 49.

And PS = 3*9 + 22 = 27+22=49, so they are equal, which matches the congruence.

So QR = 49.

Yes, that must be it. The expression "10x-41" is for QR, and "3x+22" is for PS or something, but in the diagram, likely "10x-41" is labeled on QR.

So with x=9, QR = 49.

✔ So answer is 49.

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Problem 7: Find m∠ABD

Diagram shows quadrilateral ABCD with right angles at A and C, and BD is a diagonal. Also, angle at B is split into two parts: (9x - 22)° and (6x - 7)°, and there are arc marks indicating that BD is the angle bisector of angle ABC.

Also, since angles at A and C are right angles, and BD is angle bisector, then by the angle bisector theorem or properties, the perpendicular distances from D to the sides are equal, but here it's from B.

Actually, similar to problem 6.

BD is the angle bisector of angle ABC.

From point D, we have perpendiculars to BA and BC? Angles at A and C are right angles, so DA ⊥ BA, and DC ⊥ BC.

So, D is on the angle bisector of angle ABC, so the perpendicular distances from D to the two sides BA and BC are equal.

The perpendicular distance to BA is DA, and to BC is DC.

So DA = DC.

But in the diagram, no expressions for DA or DC.

Instead, we have angles at B: angle ABD = (9x - 22)°, angle CBD = (6x - 7)°, and since BD is angle bisector, these two angles are equal.

Is that correct? If BD bisects angle ABC, then yes, angle ABD = angle CBD.

So set them equal:

9x - 22 = 6x - 7

Subtract 6x:

3x - 22 = -7

Add 22:

3x = 15

x = 5

Now, m∠ABD = 9x - 22 = 9*5 - 22 = 45 - 22 = 23°

Check other angle: 6*5 - 7 = 30 - 7 = 23° → equal, good.

✔ Correct.

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Now, compiling all answers:

1. Perimeter of ΔABC = 70
2. MN = 19
3. m∠PST = 129°
4. x = 13
5. JL = 38
6. QR = 49
7. m∠ABD = 23°

Final Answer:
1. 70
2. 19
3. 129
4. 13
5. 38
6. 49
7. 23