The shaded rectangle is tangent to the circle at the two points indicated, and its bottom right corner lies on the circumference of that circle. What is the area of the rectangle?
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Like with many geometry problems, it is helpful to draw a few auxiliary lines:
- At the tangent point on the left, we draw a horizontal line across, parallel to the top side of the rectangle.
- At the tangent point on the top, we draw a vertical line down, parallel to the left side of the rectangle.
- From the center of the circle, draw a radius to the bottom right corner of the rectangle.
The two tangent lines and the two radii drawn from the tangent points to the center of the circle form a square. We know this is a square because:
- Its angles are right angles (the top left corner is a right angle because it is part of the rectangle, the bottom left and top right corners are right angles because radii always form right angles to the tangent lines at the tangent point), so it must be a rectangle
- The bottom and right sides are equal because they are both radii, and a rectangle with adjacent sides equal is a square
This tells us the radius of the circle is 5.
Now we know two of the three sides of the right triangle we have drawn in the bottom right. Using the Pythagorean theorem, we know the remaining side must be 3.
Therefore the sides of the rectangle are 5 + 4 = 9 and 5 + 3 = 8, which gives us an area of 72.