A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adj

Q: Why is the center of the rectangle the same as the center of the circle?

Because the rectangle is inscribed, its vertices lie on the circle. The symmetry of the rectangle forces its center to be the center of the circle.

Q: Why is the perpendicular bisector of a horizontal line vertical?

A horizontal line has a slope of $0$. The perpendicular slope is undefined, which corresponds to a vertical line.

Q: Can we use the distance formula for the height?

Yes, but it is overkill. The vertical distance is simply the difference in $y$-coordinates between the center and the side.

Question

A rectangle is inscribed in a circle with a diameter lying along the line $3y = x + 7$. If the two adjacent vertices of the rectangle are $(-8, 5)$ and $(6, 5)$, then the area of the rectangle (in sq. units) is :-

Options:

(A) $72$

(B) $84$

(C) $98$

(D) $56$

Sigma Percentile · Accepted Answer

### Analyzing the Horizontal Side

Consider the points $A(-8, 5)$ and $B(6, 5)$. Since both points share the same $y$-coordinate of $5$, the side $AB$ is perfectly horizontal.

The length of side $AB$ is calculated by the difference in the $x$-coordinates:
$$|6 - (-8)| = 14 	ext{ units}$$

This value represents the length of our rectangle. We will utilize this dimension for the final area calculation.

### The Symmetry of the Circle

A rectangle inscribed in a circle shares the same center as the circle. By geometric symmetry, the center must lie on the perpendicular bisector of any side.

The midpoint $M$ of side $AB$ is calculated as:
$$M = \left( \frac{-8 + 6}{2}, 5 ight) = (-1, 5)$$

Since $AB$ is horizontal, its perpendicular bisector is a vertical line passing through $x = -1$. Thus, the center of the rectangle must lie on the line $x = -1$.

### The Intersection of Truth

The problem provides the equation of a diameter: $3y = x + 7$. Because the center of the circle must lie on all diameters, it must satisfy this equation.

We find the center by substituting $x = -1$ into the diameter equation:
$$3y = -1 + 7$$
$$3y = 6$$
$$y = 2$$

The center of the circle and the rectangle is located at the point $C(-1, 2)$.

### The Final Calculation

The vertical distance from the center $C$ (at $y = 2$) to the top side $AB$ (at $y = 5$) is:
$$5 - 2 = 3 	ext{ units}$$

Since the center is the midpoint of the rectangle, this distance represents half of the total height. Therefore, the total height of the rectangle is:
$$2 	imes 3 = 6 	ext{ units}$$

The area of the rectangle is the product of its length and height:
$$	ext{Area} = 14 	imes 6 = 84 	ext{ square units}$$

The final answer is **84**.

Animated Solution for Mathematics - Circles: A rectangle is inscribed in a circle with a diameter lying along the line $3 y = x + 7$ . If the two adjacent vertices of the rectangle are $(- 8, 5)$ and $(6, 5)$ , then the area of the rectangle (in sq. units) is :-

Select Answer:

Visualized Solution

Visualize the Given Points

Calculate Side Length $A B$

Symmetry and the Center

Equation of the Perpendicular Bisector

Intersecting with the Diameter

Calculate the Center Coordinates

Determine the Rectangle's Height

Calculate the Final Area

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Horizontal Side

The Symmetry of the Circle

The Intersection of Truth

The Final Calculation

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