https://file.aiquickdraw.com/mathbot/file/1726261828047biymncjn.jpg

Let's tackle each of the two problems shown in the image.

### Problem 9: Classify a triangle with side lengths 15, 11, and 12
To classify the triangle, we need to check whether it is:
1. **Scalene, isosceles, or equilateral** (based on side lengths).
2. **Acute, obtuse, or right-angled** (based on angles).

#### Step 1: Check for type based on side lengths
- **Scalene**: A triangle is scalene if all three sides have different lengths. In this case, 15, 11, and 12 are all different, so the triangle is **scalene**.

#### Step 2: Check for angle type using the Pythagorean theorem
We can check if the triangle is a **right triangle** using the Pythagorean theorem, \( c^2 = a^2 + b^2 \), where \( c \) is the longest side (15 in this case).

\[
c^2 = 15^2 = 225 \quad \text{and} \quad a^2 + b^2 = 11^2 + 12^2 = 121 + 144 = 265
\]
Since \( 225 \neq 265 \), this is **not a right triangle**.

#### Step 3: Check if it is acute or obtuse
We can check if the triangle is **acute** (all angles less than 90°) or **obtuse** (one angle greater than 90°) by comparing:
- If \( c^2 > a^2 + b^2 \), the triangle is **obtuse**.
- If \( c^2 < a^2 + b^2 \), the triangle is **acute**.

In this case, since \( c^2 = 225 \) and \( a^2 + b^2 = 265 \), \( c^2 < a^2 + b^2 \), so the triangle is **acute**.

### Conclusion: The triangle is a **scalene acute triangle**.

---

### Problem 10: Give reasons why the following triangle is not possible
Looking at the given triangle with side lengths 21, 28, and 35 and angles \( 47^\circ \), \( 33^\circ \), and a right angle (90°), here are some reasons why this triangle is not possible:

1. **Sum of angles**: The angles of any triangle must add up to \( 180^\circ \). In this case, the angles given are \( 47^\circ \), \( 33^\circ \), and \( 90^\circ \), which sum to:
   \[
   47^\circ + 33^\circ + 90^\circ = 170^\circ
   \]
   This is incorrect because the sum should be exactly \( 180^\circ \).

2. **Angle-side relationship**: In a right triangle, the hypotenuse (longest side) should be opposite the \( 90^\circ \) angle. Here, the longest side is 35, but it is not positioned opposite the right angle. Instead, 21 is adjacent to the right angle, which violates the fundamental properties of a right triangle.

3. **Invalid Pythagorean relationship**: In a right triangle, the Pythagorean theorem should hold, meaning \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Here, the sides are 21, 28, and 35. Checking the Pythagorean relationship:
   \[
   21^2 + 28^2 = 441 + 784 = 1225
   \]
   \[
   35^2 = 1225
   \]
   The equation does hold, so the side lengths satisfy the Pythagorean theorem. However, the placement of sides with respect to angles is incorrect.

4. **Mismatch between angles and side lengths**: The triangle's sides and angles do not align correctly, making it geometrically impossible.

### Conclusion: The triangle is not possible due to inconsistencies between its angles and side placement.

---

Do you have any questions or would you like more details on either of these problems?

Here are 5 related questions to explore further:
1. How do you classify triangles based on side lengths and angles?
2. What is the difference between a right triangle and an acute triangle?
3. How can you verify if a triangle is obtuse?
4. Why is the sum of angles in a triangle always \( 180^\circ \)?
5. How does the Pythagorean theorem apply to real-life triangle problems?

**Tip**: Always verify both the side lengths and angles when determining if a triangle is possible and properly classified.

Classify a triangle with side lengths 15, 11, 12. Give as many reasons as you can as to why the following triangle is not possible (triangle with angles 47°, 33°, and 90°, and sides 21, 28, and 35).

Learn how to classify a triangle with sides 15, 11, 12, and understand why a triangle with sides 21, 28, 35 and angles 47°, 33°, and 90° is not geometrically possible. This problem covers triangle classification, the Pythagorean theorem, and angle-sum properties. Suitable for Grades 7-9.

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