Solved Name: Unit 6: Similar Triangles Date: Bell: Homework ... - Free Printable

Let's solve each problem step by step from your homework on similar triangles and similar figures.

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Problem 1: ΔFGH ~ ΔKJI

We are told that triangle $ \triangle FGH \sim \triangle KJI $. This means the triangles are similar, so their corresponding angles are congruent and their corresponding sides are proportional.

#### Step 1: Identify corresponding angles
Since the triangles are named in order (F → K, G → J, H → I), we match the vertices:

- $ \angle F \cong \angle K $
- $ \angle G \cong \angle J $
- $ \angle H \cong \angle I $

#### Step 2: List corresponding sides
Using the same vertex order:
- $ FG \leftrightarrow KJ $
- $ GH \leftrightarrow JI $
- $ FH \leftrightarrow KI $

So the proportion relating corresponding sides is:

$$
\frac{FG}{KJ} = \frac{GH}{JI} = \frac{FH}{KI}
$$

#### ✔ Final Answer for Problem 1:

| Angles | Sides |
|--------------------|--------------------|
| $ \angle F \cong \angle K $ | $ FG \leftrightarrow KJ $ |
| $ \angle G \cong \angle J $ | $ GH \leftrightarrow JI $ |
| $ \angle H \cong \angle I $ | $ FH \leftrightarrow KI $ |

Proportion:
$$
\frac{FG}{KJ} = \frac{GH}{JI} = \frac{FH}{KI}
$$

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Problem 2: Scale factor of Figure A to Figure B

Given two rectangles:

- Figure A: width = 4, height = 2
- Figure B: width = 8, height = 16

We want the scale factor of A to B, which means:

$$
\text{Scale factor} = \frac{\text{Dimension of B}}{\text{Dimension of A}}
$$

Use either dimension:

- Width: $ \frac{8}{4} = 2 $
- Height: $ \frac{16}{2} = 8 $

Wait — this is not consistent! That suggests a mistake. But let’s double-check.

Wait — if the rectangles are similar, their dimensions must be proportional.

But here:
- A: 4 × 2 → ratio = 2:1
- B: 8 × 16 → ratio = 1:2 → not the same!

That can’t be right. Wait — maybe it's a typo or mislabeling?

Wait — actually, look again. If Figure B has width 8 and height 16, then its aspect ratio is $ 8:16 = 1:2 $, while Figure A is $ 4:2 = 2:1 $. So they are not similar unless we're misunderstanding.

But the problem says “the pairs of polygons below are similar.” So perhaps there’s a labeling error?

Wait — maybe the rectangle labeled B is not 8×16? Let’s recheck the image description.

From your image:
- Figure A: 4 units wide, 2 units high
- Figure B: 8 units wide, 16 units high

But these are not similar because:
- A: $ \frac{4}{2} = 2 $
- B: $ \frac{8}{16} = 0.5 $

So aspect ratios are different → not similar unless the labels are wrong.

But the problem says they are similar. So perhaps the figure is oriented differently?

Wait — maybe the height of B is not 16? Or is it?

Looking at the image:
- Figure B is drawn taller than wide, with side labeled 16 vertically and 8 horizontally.
- So yes: 8 (horizontal), 16 (vertical) → ratio = 8:16 = 1:2
- Figure A: 4 (horizontal), 2 (vertical) → ratio = 4:2 = 2:1 → not matching.

But 2:1 vs 1:2 → not similar.

Wait — unless the corresponding sides are matched properly.

Maybe the height of A is 2, and width is 4, and B has height 16, width 8 — but if we match height to height, then:

- Height: $ \frac{16}{2} = 8 $
- Width: $ \frac{8}{4} = 2 $

Still not equal → not similar.

This is a contradiction.

But wait — perhaps the figure B is 16 units wide and 8 units high? That would make sense.

Wait — looking at the image: it says "16" on the vertical side and "8" on the horizontal side. So it's taller than wide.

But Figure A is wider than tall.

So unless one is rotated, they’re not similar.

But the problem says they are similar — so maybe the sides are not labeled correctly, or we need to assume correspondence.

Alternatively, perhaps the side lengths are meant to be compared in order.

Wait — maybe Figure A is 4 and 2, and Figure B is 8 and 16, but the ratio should be constant.

Try scaling A by a factor of 2:
- 4 × 2 = 8
- 2 × 2 = 4 → but B has height 16, not 4 → doesn't work.

Try scaling A by 8:
- 4 × 8 = 32 ≠ 8 → no.

Wait — what if we scale A to get B, but the sides are reversed?

Suppose:
- The shorter side of A is 2, longer is 4
- The shorter side of B is 8, longer is 16 → no, 8 < 16

So shorter side of A: 2 → B: 8 → ratio 4
Longer side: 4 → 16 → ratio 4

Ah! Wait — if the sides are scaled by 4:

- 2 × 4 = 8
- 4 × 4 = 16

Yes! So A is 2×4, B is 8×16, so:

- $ \frac{8}{2} = 4 $
- $ \frac{16}{4} = 4 $

So the scale factor of A to B is 4

But wait — the dimensions of A are: width 4, height 2 → so the height is 2, width is 4

Then B has height 16, width 8

So:
- Height: $ \frac{16}{2} = 8 $
- Width: $ \frac{8}{4} = 2 $

Not matching.

Wait — unless the orientation is switched.

But if we consider corresponding sides, maybe:

- The vertical side of A is 2, of B is 16 → ratio $ \frac{16}{2} = 8 $
- The horizontal side of A is 4, of B is 8 → ratio $ \frac{8}{4} = 2 $

Still not equal.

This suggests they are not similar.

But the problem says they are similar. So likely, there is a mistake in my reading.

Wait — perhaps Figure B is 8 units wide and 16 units high, but Figure A is 4 units wide and 2 units high — so both have the same shape: 4:2 = 2:1, and 8:16 = 1:2 — different.

No.

Wait — 4:2 = 2:1 → A is wider than tall
8:16 = 1:2 → B is taller than wide → not same shape.

So unless the figure is flipped, they are not similar.

But the problem says they are. So perhaps the labels are swapped?

Wait — maybe Figure B has width 16 and height 8?

But in the image, it shows:
- Vertical side labeled 16
- Horizontal side labeled 8

So it's 16 units high, 8 units wide

So ratio: 16:8 = 2:1

Figure A: 2 units high, 4 units wide → ratio: 2:4 = 1:2 → still not same.

Wait — unless we compare height to height, width to width:

- Height A: 2, Height B: 16 → ratio = 8
- Width A: 4, Width B: 8 → ratio = 2

Not equal.

But if we compare A's height to B's width, etc., that’s not valid.

Wait — unless Figure A is 2 × 4, and Figure B is 8 × 16, but we’re supposed to see that:

- 2 → 8 (×4)
- 4 → 16 (×4)

But only if the 2 corresponds to 8, and 4 corresponds to 16

But in Figure A, 2 is the height, 4 is the width

In Figure B, 16 is the height, 8 is the width

So if we map:
- A's height (2) → B's height (16): ratio = 8
- A's width (4) → B's width (8): ratio = 2

Not equal.

But if we map:
- A's height (2) → B's width (8): ratio = 4
- A's width (4) → B's height (16): ratio = 4

Then both ratios are 4 — so if we rotate Figure B, then it matches.

But rotating doesn't change similarity — so if they have the same shape, they are similar.

But here, A is 2×4 → area 8
B is 8×16 → area 128

But 2×4 and 8×16 — if you scale 2×4 by 4, you get 8×16 → yes!

Wait — scale factor of 4:

- 2 × 4 = 8
- 4 × 4 = 16

So if Figure A is 2×4, and Figure B is 8×16, then yes, they are similar with scale factor 4.

But in the diagram, Figure A has height 2, width 4
Figure B has height 16, width 8

So the height of A is 2, height of B is 16 → ratio 8
Width of A is 4, width of B is 8 → ratio 2

So unless the corresponding sides are not aligned, they aren't similar.

But if we assume that the side of length 2 in A corresponds to the side of length 8 in B, and 4 corresponds to 16, then:

- 2 → 8 → ×4
- 4 → 16 → ×4

So scale factor is 4

But then the orientation is different — meaning B is scaled and possibly rotated.

But in terms of similarity, rotation is allowed.

So even though the orientation looks different, if the ratios of corresponding sides are equal, they are similar.

So the key is: which sides correspond?

The problem says the polygons are similar, so we assume the corresponding sides are in order.

But in the diagram, it's not clear how they are labeled.

But based on the numbers:

- A: 2 and 4
- B: 8 and 16

And since 2:8 = 1:4, 4:16 = 1:4 → so if the smaller side of A corresponds to the smaller side of B, and larger to larger, then:

- 2 ↔ 8 → ratio 4
- 4 ↔ 16 → ratio 4

But in Figure B, the smaller side is 8, larger is 16 → so 8 is smaller, 16 is larger

In A: 2 (smaller), 4 (larger)

So:
- 2 ↔ 8 → ratio 4
- 4 ↔ 16 → ratio 4

Yes! So scale factor of A to B is 4

Even though the diagram may show them oriented differently, as long as the ratios are preserved, they are similar.

So scale factor of A to B is $ \boxed{4} $

✔ Answer for Problem 2: 4

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Problem 3: Scale factor of Figure A to Figure B

Given two triangles:

- Triangle A: sides 10, 18, 15
- Triangle B: sides 6, 4, 7.2

We need to find the scale factor of A to B.

First, check if they are similar by seeing if the ratios of corresponding sides are equal.

Assume the sides are listed in the same order.

So:
- A: 10, 18, 15
- B: 6, 4, 7.2

Check ratios:

- $ \frac{6}{10} = 0.6 $
- $ \frac{4}{18} \approx 0.222 $
- $ \frac{7.2}{15} = 0.48 $

Not equal — so not in that order.

Try matching smallest to smallest, etc.

List sides in ascending order:

- A: 10, 15, 18
- B: 4, 6, 7.2

Now check ratios:

- $ \frac{4}{10} = 0.4 $
- $ \frac{6}{15} = 0.4 $
- $ \frac{7.2}{18} = 0.4 $

Yes! All ratios are 0.4

So the scale factor of A to B is $ \frac{1}{2.5} = 0.4 $, or $ \frac{2}{5} $

But scale factor is usually written as a fraction or decimal.

So $ \boxed{0.4} $ or $ \boxed{\frac{2}{5}} $

But since the question asks for scale factor of A to B, and A is larger, it should be less than 1.

So scale factor = 0.4

✔ Answer for Problem 3: 0.4 or $ \frac{2}{5} $

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Problem 4: Scale factor of A to B is 4:5, find x

Given:
- Triangle A has a side of length 4
- Triangle B has a corresponding side of length x
- Scale factor of A to B is 4:5

Scale factor = $ \frac{\text{A}}{\text{B}} = \frac{4}{5} $

So:
$$
\frac{4}{x} = \frac{4}{5}
$$

Cross-multiply:
$$
4 \cdot 5 = 4 \cdot x \Rightarrow 20 = 4x \Rightarrow x = 5
$$

✔ Answer: $ x = 5 $

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Problem 5: Scale factor of A to B is 7:2, find perimeter of A

Given:
- Triangle B has sides: 9, 12, 5
- So perimeter of B = 9 + 12 + 5 = 26
- Scale factor of A to B is 7:2 → meaning $ \frac{\text{A}}{\text{B}} = \frac{7}{2} $

So perimeter of A = $ \frac{7}{2} \times \text{perimeter of B} = \frac{7}{2} \times 26 = 7 \times 13 = 91 $

✔ Answer: Perimeter of A = 91

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✔ Final Answers:

1. Angles:
- $ \angle F \cong \angle K $
- $ \angle G \cong \angle J $
- $ \angle H \cong \angle I $
Sides:
- $ FG \leftrightarrow KJ $
- $ GH \leftrightarrow JI $
- $ FH \leftrightarrow KI $
Proportion: $ \frac{FG}{KJ} = \frac{GH}{JI} = \frac{FH}{KI} $

2. Scale factor of A to B: $ \boxed{4} $

3. Scale factor of A to B: $ \boxed{0.4} $ or $ \boxed{\frac{2}{5}} $

4. Value of x: $ \boxed{5} $

5. Perimeter of A: $ \boxed{91} $

Let me know if you'd like a visual explanation or help with any part!