Two 45-45-90 right triangles with given side lengths and angles.

It looks like you've uploaded an image with two right triangles, and I'll solve the problem based on the information provided.

Let’s analyze each triangle:

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Triangle ABT (Left Triangle)

- Right angle at B → ∠B = 90°
- Angle at T = 45°
- Side AB = 8 (this is one leg of the right triangle)
- We are to find other sides or angles if needed.

Since it's a right triangle with a 45° angle, and one angle is 90°, the third angle must be:
> ∠A = 180° – 90° – 45° = 45°

So this is a 45°-45°-90° triangle, which means it's an isosceles right triangle — the two legs are equal.

Given:
- AB = 8 (one leg)
- Since both legs are equal: BT = AB = 8

Now, use the 45°-45°-90° triangle rule:
> Hypotenuse = leg × √2

So:
> AT = 8√2

✔ So for triangle ABT:
- AB = 8
- BT = 8
- AT = 8√2

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Triangle OBY (Right Triangle on the Right)

- Right angle at B → ∠B = 90°
- ∠O = 45°
- ∠Y = 45°
- Hypotenuse OY = 10

Again, this is a 45°-45°-90° triangle.

In such triangles:
- The legs are equal
- Each leg = hypotenuse / √2

So:
> OB = BY = 10 / √2 = (10√2) / 2 = 5√2

✔ So for triangle OBY:
- OB = 5√2
- BY = 5√2
- OY = 10

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✔ Summary of Results

#### Triangle ABT:
- AB = 8
- BT = 8
- AT = 8√2
- Angles: ∠A = 45°, ∠B = 90°, ∠T = 45°

#### Triangle OBY:
- OB = 5√2
- BY = 5√2
- OY = 10
- Angles: ∠O = 45°, ∠B = 90°, ∠Y = 45°

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🔍 Possible Task Interpretation

Since no explicit question was given, possible tasks might include:

1. Find missing side lengths → Done above.
2. Compare the two triangles → Both are 45°-45°-90° right triangles.
3. Find ratios or scale factor → Let's check that.

Compare corresponding sides:

- AB = 8, OB = 5√2 ≈ 7.07 → not the same
- But notice: ABT has leg 8, OBY has leg 5√2 ≈ 7.07 → different sizes

But since both are 45°-45°-90°, they are similar (same shape, different size).

Scale factor from OBY to ABT:
> 8 / (5√2) = 8 / (5×1.414) ≈ 8 / 7.07 ≈ 1.13

Or rationalized:
> 8 / (5√2) = (8√2)/10 = (4√2)/5

So the ratio of ABT to OBY is (4√2)/5.

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🟩 Final Answer:

Both triangles are 45°-45°-90° right triangles.

- In triangle ABT:
- Legs: AB = BT = 8
- Hypotenuse: AT = $ 8\sqrt{2} $

- In triangle OBY:
- Legs: OB = BY = $ 5\sqrt{2} $
- Hypotenuse: OY = 10

✔ These triangles are similar but not congruent.

If your original task was to find missing sides or verify properties, this is the complete solution.

Let me know if there was a specific question (e.g., "Are the triangles congruent?", "Find the area", etc.), and I can tailor the answer!