The problem shown in the image asks for the area of an equilateral triangle with a center point $B$. From the image, we are provided the following information:

• The radius of the inscribed circle (distance from center $B$ to the midpoint of a side) is 7 inches.
• The distance from the center $B$ to one of the vertices is 10 inches.

Solving the Problem:

This triangle is equilateral, meaning all sides and angles are equal. The key to finding the area is to first calculate the side length of the equilateral triangle using the relationship between the circumcenter (the center of the equilateral triangle) and its vertices.

Step 1: Side Length of the Equilateral Triangle

In an equilateral triangle, the distance from the center to a vertex (denoted as 10 inches) is known as the circumradius $R$, while the distance from the center to the midpoint of a side (denoted as 7 inches) is known as the inradius $r$.

For an equilateral triangle, the side length $s$ can be related to the circumradius $R$ using the following formula: $s = \frac{R}{\sin(60^\circ)} = R \times \frac{2}{\sqrt{3}}$

Substituting $R = 10$ inches: $s = 10 \times \frac{2}{\sqrt{3}} \approx 11.55 \, \text{inches}$

Step 2: Area of the Equilateral Triangle

The area $A$ of an equilateral triangle with side length $s$ is given by the formula: $A = \frac{s^2 \sqrt{3}}{4}$

Substituting $s = 11.55$: $A = \frac{(11.55)^2 \sqrt{3}}{4} \approx \frac{133.43 \times 1.732}{4} \approx 57.79 \, \text{square inches}$

So, the area of the equilateral triangle is approximately 57.8 square inches when rounded to the nearest tenth.

Would you like further details on any part of the solution?

Here are five related questions to extend your understanding:

1. What is the relationship between the circumradius and the inradius in an equilateral triangle?
2. How can we calculate the height of an equilateral triangle given the side length?
3. Why does an equilateral triangle always have equal angles?
4. What is the formula for the area of an equilateral triangle in terms of its inradius?
5. How would the area change if the circumradius were increased by 1 inch?

Tip: In geometry, understanding the properties of triangles, especially regular ones like equilateral triangles, can simplify many calculations and reveal useful patterns for solving other problems.