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The problem describes a thin triangular plate with vertices at \( v_1 = (1,3) \), \( v_2 = (9,4) \), and \( v_3 = (7,6) \), and it asks to complete parts (a) and (b) below. The plate has uniform density and thickness, and the mass of the plate is given as 3 g.

Let's solve part by part. Usually, such problems ask for the following:

1. **Centroid of the triangle**: The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be found using the formula:
   \[
   G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
   \]
   We can calculate the coordinates of the centroid.

2. **Moment of inertia or mass distribution**: If this is part of the question, the mass of the plate might be used to find moments about different axes.

Let's start with part (a) by finding the centroid and then tackle part (b) if more information becomes clear.

### Part (a): Centroid Calculation

Given the vertices:
- \( v_1 = (1, 3) \)
- \( v_2 = (9, 4) \)
- \( v_3 = (7, 6) \)

We use the formula for the centroid:

\[
G = \left( \frac{1 + 9 + 7}{3}, \frac{3 + 4 + 6}{3} \right)
\]

Let's calculate that.The centroid \( G \) of the triangular plate is located at \( \left( 5.67, 4.33 \right) \).

### Part (b) likely involves either calculating the moment of inertia or another physical property of the triangular plate, but without more specific details in the image, we can only speculate. If the moment of inertia or mass distribution is part of the question, please let me know.

Would you like to clarify the details of part (b) or go over the explanation in more depth? Here are some related questions you might explore:

1. How would you find the area of this triangle?
2. What is the moment of inertia of this triangular plate about the x-axis?
3. What is the moment of inertia of this plate about the y-axis?
4. How does the mass of the plate affect its center of mass?
5. How would the centroid change if the plate had varying density?

**Tip**: The centroid of a triangle always divides each median into two segments, where one segment is twice the length of the other.

A thin triangular plate of uniform density and thickness has vertices at v₁ = (1,3), v₂ = (9,4), v₃ = (7,6), and the mass of the plate is 3 g. Complete parts a and b below.

Learn how to calculate the centroid of a triangular plate with vertices at (1,3), (9,4), and (7,6), using the centroid formula. Explore related topics in geometry, mass distribution, and centroid properties.

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