Unit 19 Section 2 : Similar shapes - Free Printable

It looks like you've uploaded an image with two right triangles, and we are to solve a problem involving them. Let's analyze the given information.

Given:

- Triangle ABC is a right triangle with:
- ∠A = 90° (right angle)
- AB = ? (unknown)
- AC = 12 cm
- BC = 13 cm
- Triangle DEF is a right triangle with:
- ∠D = 90° (right angle)
- DE = 2.5 cm
- DF = 6 cm
- EF = ? (unknown)

We are likely being asked to find missing side lengths or possibly determine if the triangles are similar or congruent. Since both are right triangles, let’s proceed step by step.

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Step 1: Solve for missing side in Triangle ABC

In triangle ABC:
- Right-angled at A.
- AC = 12 cm (one leg)
- BC = 13 cm (hypotenuse)
- AB = ? (other leg)

Use the Pythagorean Theorem:

$$
AB^2 + AC^2 = BC^2
$$

$$
AB^2 + 12^2 = 13^2
$$

$$
AB^2 + 144 = 169
$$

$$
AB^2 = 169 - 144 = 25
$$

$$
AB = \sqrt{25} = 5\ \text{cm}
$$

✔ So, AB = 5 cm

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Step 2: Solve for missing side in Triangle DEF

In triangle DEF:
- Right-angled at D.
- DE = 2.5 cm (one leg)
- DF = 6 cm (other leg)
- EF = ? (hypotenuse)

Again, use the Pythagorean Theorem:

$$
EF^2 = DE^2 + DF^2
$$

$$
EF^2 = (2.5)^2 + 6^2 = 6.25 + 36 = 42.25
$$

$$
EF = \sqrt{42.25} = 6.5\ \text{cm}
$$

✔ So, EF = 6.5 cm

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Step 3: Compare the Triangles

Now let's look at the sides:

#### Triangle ABC:
- AB = 5 cm
- AC = 12 cm
- BC = 13 cm

#### Triangle DEF:
- DE = 2.5 cm
- DF = 6 cm
- EF = 6.5 cm

Let’s check if the sides are proportional:

Compare ratios:

- AB / DE = 5 / 2.5 = 2
- AC / DF = 12 / 6 = 2
- BC / EF = 13 / 6.5 = 2

✔ All ratios are equal to 2, so the triangles are similar with a scale factor of 2.

Also, since both are right triangles and their corresponding sides are in proportion, they are similar.

Moreover, triangle ABC has sides 5, 12, 13 — a well-known Pythagorean triple.

Triangle DEF has sides 2.5, 6, 6.5 — which is half of 5, 12, 13? Wait:

Wait:
2.5 × 2 = 5
6 × 2 = 12
6.5 × 2 = 13 ✔

So, triangle DEF is half the size of triangle ABC.

Therefore, triangle DEF is a scaled-down version of triangle ABC with scale factor $ \frac{1}{2} $.

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Final Answers:

1. AB = 5 cm
2. EF = 6.5 cm
3. The triangles are similar with a scale factor of $ \frac{1}{2} $

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If your original task was to find the missing side(s), then:

- AB = 5 cm
- EF = 6.5 cm

Let me know if there was a specific question (e.g., "Are the triangles similar?", "Find the area", etc.), but based on the diagram, these are the most logical conclusions.