Quiz & Worksheet - Applications of Similar Triangles | Study.com - Free Printable

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

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Problem 1:

We are told that triangle HYV and triangle AYB are similar by AA similarity. That means their corresponding sides are proportional.

Looking at the diagram:

- In triangle AYB: side AY = x + 17? Wait — actually, looking again: point A is on HY, and YA is labeled as 17, and HA is labeled as x. So total HY = x + 17.
- Side YB = 16
- Side BV = 22 → so YV = YB + BV = 16 + 22 = 38

Since triangles AYB ~ HYV (note order: A corresponds to H, Y to Y, B to V), then:

Corresponding sides:
AY / HY = YB / YV

So:

17 / (x + 17) = 16 / 38

Now cross-multiply:

17 * 38 = 16 * (x + 17)

Calculate 17 * 38:

17 * 38 = (10*38) + (7*38) = 380 + 266 = 646

So:

646 = 16(x + 17)

Divide both sides by 16:

646 ÷ 16 = x + 17

646 ÷ 16 = 40.375

So:

40.375 = x + 17

Subtract 17:

x = 40.375 - 17 = 23.375

Wait — but 23.375 is not among the options. Let me double-check.

Hold on — maybe I misassigned the correspondence.

The problem says: Triangle HYV and triangle AYB are similar.

Order matters: H ↔ A, Y ↔ Y, V ↔ B? Or H ↔ A, Y ↔ Y, V ↔ B?

Actually, since angle at Y is common, and angles at B and V are marked equal (both have same arc marks), so angle AYB = angle HYV (same angle), and angle ABY = angle HVY (marked equal). So yes, triangle AYB ~ triangle HYV with correspondence A→H, Y→Y, B→V.

So sides:

AY corresponds to HY

YB corresponds to YV

AB corresponds to HV

So ratio: AY/HY = YB/YV

AY = 17, HY = x + 17, YB = 16, YV = 16 + 22 = 38

So 17/(x+17) = 16/38

Cross multiply: 17*38 = 16*(x+17)

17*38: let's compute again: 17*40=680, minus 17*2=34 → 680-34=646 ✔️

16*(x+17)=646 → x+17=646/16=40.375 → x=23.375

But 23.375 is not an option. Options are: 40.375, 23, 40, 29.363

Wait — perhaps I got the correspondence wrong.

Maybe triangle HYV ~ triangle AYB means H↔A, Y↔Y, V↔B — which is what I did.

Alternatively, maybe it’s triangle HYV ~ triangle AYB meaning H↔A, Y↔Y, V↔B — same thing.

Perhaps the side labeled “x” is not HA but something else? Looking back at the diagram description: “x” is written next to segment HA, and “17” next to AY, so HY = HA + AY = x + 17. That seems right.

Wait — maybe the proportion is set up differently.

Another way: since the small triangle is inside the big one, and they share angle Y, and base angles are equal, so the ratios should be:

Small triangle side / Big triangle side = same ratio.

So for side from Y: in small triangle, YB = 16; in big triangle, YV = 38.

So scale factor = 16/38 = 8/19

Then, side AY in small triangle corresponds to side HY in big triangle.

So AY / HY = 8/19

17 / (x + 17) = 8/19

Cross multiply: 17 * 19 = 8 * (x + 17)

17*19 = 323

8*(x+17) = 323

x+17 = 323/8 = 40.375

x = 40.375 - 17 = 23.375 — still same.

But 23.375 isn't an option. Closest is 23 or 29.363.

Wait — perhaps "x" is the entire length HY? But the diagram shows "x" on HA and "17" on AY, so likely x is HA.

Maybe the labeling is different. Let me read the problem again: “Triangle HYV and triangle AYB are similar”

Points: H, Y, V form the big triangle. A is on HY, B is on YV.

So triangle AYB has points A, Y, B.

Similar to HYV.

So vertex A corresponds to H, Y to Y, B to V.

So side AY corresponds to HY? No — in similar triangles, corresponding sides are between corresponding vertices.

So side AY is between A and Y, which correspond to H and Y, so side AY corresponds to side HY.

Yes.

Side YB corresponds to YV.

Side AB corresponds to HV.

So ratio AY/HY = YB/YV

17 / (x + 17) = 16 / 38

As before.

But 16/38 reduces to 8/19, and 17 / (x+17) = 8/19

Then x+17 = 17 * 19 / 8 = 323 / 8 = 40.375

x = 23.375

Not in options. Perhaps there's a mistake in the problem or my interpretation.

Wait — look at the options: 40.375 is there. Maybe they want x+17? But the question asks for x.

Or perhaps "x" is labeled on the whole HY? But the diagram shows "x" on HA and "17" on AY, so x is part of it.

Another possibility: maybe the 17 is not AY but something else. Let me assume that "x" is the length from H to A, and "17" is from A to Y, so HY = x + 17.

Perhaps the proportion is inverted.

Suppose we do HY / AY = YV / YB

Then (x+17)/17 = 38/16

38/16 = 19/8 = 2.375

So x+17 = 17 * 2.375 = let's calculate: 17*2 = 34, 17*0.375=6.375, total 40.375

So x = 40.375 - 17 = 23.375 again.

Same result.

But 23.375 is not an option. Option 2 is 23, close but not exact.

Option 4 is 29.363 — how would that come?

Perhaps if I used different correspondence.

Suppose triangle HYV ~ triangle AYB with H↔A, Y↔B, V↔Y — but that doesn't make sense because angle at Y is common.

Another idea: perhaps the side labeled "17" is not AY but AB or something, but the diagram description says "17" is on AY.

Let's look at the numbers: 16 and 22 on the base, so YV = 38.

If I set up proportion as AY / YB = HY / YV

17 / 16 = (x+17) / 38

Then 17/16 = (x+17)/38

Cross multiply: 17*38 = 16*(x+17) — same as before.

I think there might be a typo in the options, or in my understanding.

Wait — perhaps "x" is the length of HY, not HA. Let me check the diagram description: "x" is written next to the segment from H to A, and "17" from A to Y, so x is HA.

But let's try assuming x is HY. Then if x = HY, and AY = 17, then HA = x - 17.

Then proportion: AY / HY = YB / YV => 17/x = 16/38

Then 17/x = 8/19

x = 17 * 19 / 8 = 323/8 = 40.375

And 40.375 is an option! And the question asks for x, which would be HY.

In the diagram, "x" is labeled on the segment from H to A, but perhaps it's a mislabel, or perhaps in some interpretations, x is the whole thing.

Given that 40.375 is an option, and it matches when x = HY, and the calculation gives that, probably that's what is intended.

Perhaps the "x" is meant to be the length from H to Y, and the "17" is part of it, but the label is placed on HA by mistake.

In many such problems, x is the unknown side of the larger triangle.

So let's go with that.

So if x = HY, then from similarity:

AY / HY = YB / YV

17 / x = 16 / 38

16x = 17 * 38 = 646

x = 646 / 16 = 40.375

Yes, and that's option 1.

So likely, despite the diagram showing "x" on HA, it's intended to be on HY, or perhaps it's a common convention.

So for Problem 1, answer is 40.375.

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Problem 2:

Given triangles DAR and KMR. Find y.

Diagram: triangle DAR, with K on DR, M on AR, and KM parallel to DA? Angles are marked: at D and K have same mark, at A and M have same mark, so probably KM || DA, so triangles KMR ~ DAR.

Vertices: D, A, R for large triangle.

K on DR, M on AR, and KM connects them.

Angles: angle at D and angle at K are both marked with single arc, so equal.

Angle at A and angle at M are both marked with double arc, so equal.

So triangle KMR ~ triangle DAR, with correspondence K↔D, M↔A, R↔R.

So sides: KR corresponds to DR, MR corresponds to AR, KM corresponds to DA.

Given lengths:

DK = 11, so if K is on DR, and DK=11, KR=y, so DR = DK + KR = 11 + y

AR = AM + MR = 16 + 14 = 30? Wait, AM=16, MR=14, so AR=30

In triangle KMR, side MR = 14

In triangle DAR, side AR = 30

Since KMR ~ DAR, and R is common, so side MR corresponds to AR.

Ratio of similarity = MR / AR = 14 / 30 = 7/15

Now, side KR corresponds to DR.

KR = y, DR = 11 + y

So KR / DR = 7/15

y / (11 + y) = 7/15

Cross multiply: 15y = 7(11 + y)

15y = 77 + 7y

15y - 7y = 77

8y = 77

y = 77/8 = 9.625

Which is option 1.

Perfect.

So y = 9.625

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Problem 3:

Find x, distance between T and E.

Diagram: triangle FTV, with C on FV, E on TV, and CE drawn.

Angles: at F and C have same mark (single arc), at T and E have same mark (double arc), so probably CE || FT.

Triangle FTV, points: F top, T left, V right.

C on FV, E on TV.

Angle at F and angle at C are both marked with single arc — so angle F = angle FCE? Angle at C in triangle C EV or what.

Actually, angle at F in triangle FTV, and angle at C in triangle CEV — if CE || FT, then corresponding angles equal.

Assume CE || FT.

Then triangle CEV ~ triangle FTV.

Because angle at V common, and angle at C = angle at F (corresponding), angle at E = angle at T (corresponding).

So triangle CEV ~ triangle FTV.

Correspondence: C↔F, E↔T, V↔V.

Sides: CV corresponds to FV, EV corresponds to TV, CE corresponds to FT.

Given: FC = 6, CV = 10, so FV = FC + CV = 6 + 10 = 16

TV = TE + EV = x + ? Wait, E is on TV, and TE = x, EV = ? The diagram shows "15" under TE and EV together? Look: "x" is between T and E, and "15" is written below the whole TV, so TV = 15.

It says: "x" is distance between T and E, and below TV it says "15", so TV = 15.

Also, EV is part of it, but not given directly.

In triangle CEV and FTV:

CV / FV = EV / TV

CV = 10, FV = 16, TV = 15, EV = ?

Since TV = TE + EV = x + EV = 15, so EV = 15 - x

Now, ratio: CV / FV = EV / TV

10 / 16 = (15 - x) / 15

Simplify 10/16 = 5/8

So 5/8 = (15 - x)/15

Cross multiply: 5 * 15 = 8 * (15 - x)

75 = 120 - 8x

8x = 120 - 75 = 45

x = 45 / 8 = 5.625

Which is option 1.

Perfect.

So x = 5.625

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Now, summarizing:

Problem 1: x = 40.375 (assuming x is HY, not HA)

Problem 2: y = 9.625

Problem 3: x = 5.625

All match options.

For Problem 1, even though the diagram might suggest x is HA, the math works out only if x is HY, and 40.375 is an option, so we'll go with that.

Final Answers:

1. 40.375

2. 9.625

3. 5.625

But the user asked to provide final answer after reasoning, and since there are three problems, probably list all.

The instruction says "Final Answer: [Provide only the final correct answer]", but there are three questions.

Probably need to provide all three.

Since it's a worksheet with multiple questions, I'll list them.

Final Answer:
1. 40.375
2. 9.625
3. 5.625