SOLUTION: The diagonals of rhombus have length 4 and 6 inches and a circle is inscribed in it. Find the area of the region remaining in the rhombus not occupied by the circle.

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Question 1081669: The diagonals of rhombus have length 4 and 6 inches and a circle is inscribed in it. Find the area of the region remaining in the rhombus not occupied by the circle.
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

Since for the purposes of this problem, the rotation of the rhombus is arbitrary, plot the points (-3, 0), (0, 2), (3, 0), and (0, -2). Then create the line segments (-3, 0) to (0, 2), (0, 2) to (3, 0), and so on to create the given rhombus. This rhombus will have a center (point of intersection of the diagonals) at the origin, and therefore the origin will also be the center of the inscribed circle.

The area of the rhombus is trivial, being 4 times the area of one of the right triangles formed by any two adjacent vertices of the rhombus and the origin.

The only remaining parameter needed is the radius of the inscribed circle. By definition, an inscribed circle is tangent to each of the sides of the circumscribed polygon and the desired radius is the distance from the center of the circle to one of the points of tangency. (See figure)

Using the Two-Point form, we derive an equation for the line containing the segment AB:

Now, since a radius at a point of tangency is perpendicular to the tangent line, the slope of the line containing segment OT must be the negative reciprocal of the slope of the line tangent to point T. Using the Point-Slope form, we derive an equation of the line containing the segment OT:

Having two expressions in that are both equal to , we can solve for the -coordinate of the point of intersection, namely, point T.

(I leave the algebra steps as an exercise for the student)

Then, for the -coordinate of point T:

The measure of the segment OT is the distance between point O and point T. Using the distance formula:

(You can verify the arithmetic for yourself:

But since the formula for the area of the circle requires the radius squared, we can dispense with taking the square root of that very ugly fraction:

Hence, the area of the circle is

Since the area of the entire rhombus is , the area of the rhombus exceeds the area of the inscribed circle by the difference in their areas, namely:

Which is the exact answer. You can use your calculator to determine whatever approximation you desire.

John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52800) (Show Source):
You can put this solution on YOUR website!
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The diagonals of rhombus have length 4 and 6 inches and a circle is inscribed in it.
Find the area of the region remaining in the rhombus not occupied by the circle.
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The diagonals divide the rhombus in 4 congruent right-angled triangles. The side of the rhombus is = = . The area of each small right-angled triangle is equal to A = = 3. Using the side of the rhombus as the hypotenuse of the small triangle and the radius of the inscribed circle as the altitude, you can write the same area as S = 3 = . It gives you r = . Then the area of the circle is = squared inches. The area of the rhombus is 4 times the area of the small right-angled triangle, i.e. 4*3 = 12 squared inches. Finally, the area under the question is square inches.

Solved.