"Find the area of a rhombus with sides of length 10 in. and longer diagonal of length 16 in."

and

Also, how do you solve the problem "Find the area of a regular decagon with radius 4 cm."

1. The diagonal divides the rhombus into 2 congruent triangles. The altitude bisects the diagonal.

(16/2)^2 + Y^2 = 10^2
Y^2 = 100-64 = 36
Y = 6 = Altitude.

Area = 2(16/2 * 6) = 96 in^2.

Why did the rhombus go to the gym? Because it wanted to work on its area-cise!

To find the area of a rhombus, you can use the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals.

In this case, you're given that the longer diagonal (d1) is 16 in, and the shorter diagonal (d2) is unknown.

But wait, the rhombus has something to say! It kindly reveals that the diagonals of a rhombus are perpendicular bisectors of each other.

So, with Pythagorean theorem, we can find out that d2 = 2√(10² - 8²) = 12 in.

Now we can calculate the area: Area = (16 * 12) / 2 = 96 in². Tada!

Now, let's move on to the decagon conundrum.

Why was the regular decagon so stressed? Because it couldn't figure out how to find its area without going in circles!

To find the area of a regular decagon, you can use the formula: Area = (5/2) * r² * tan(π/10), where r is the radius.

In this case, you're given that the radius is 4 cm. Let's plug it in: Area = (5/2) * (4 cm)² * tan(π/10).

Now, this is where the circus of mathematics comes in! You need to remember that the value of tan(π/10) is approximately equal to 0.36397.

So, let's calculate the area: Area ≈ (5/2) * (4 cm)² * 0.36397 ≈ 14.5594 cm².

There you have it, the area of the regular decagon is approximately 14.5594 cm². Go celebrate with some geometrically delicious pi(e)!

Remember, always approach math problems with a sense of humor. It makes the calculations, no matter how complex, a bit more enjoyable!

To find the area of a rhombus, you can use the formula:

Area = (1/2) * d1 * d2

where d1 and d2 are the lengths of the diagonals of the rhombus.

In this case, you are given the side length (10 in.) and the length of one diagonal (16 in.).

To find the length of the other diagonal, you can use the properties of a rhombus. Since the diagonals of a rhombus are perpendicular bisectors of each other, they divide the rhombus into four congruent right triangle. Using the Pythagorean theorem, you can find the length of the other diagonal.

Let's call the other diagonal d2. Since the side length of the rhombus is 10 in., the hypotenuse of each right triangle is 10 in. and the length of one leg is half of the side length (5 in.). Using the Pythagorean theorem:

d2^2 = 10^2 - 5^2
d2^2 = 100 - 25
d2^2 = 75
d2 = √75
d2 ≈ 8.66 in.

Now that you have the lengths of both diagonals (d1 = 16 in. and d2 ≈ 8.66 in.), you can plug them into the area formula:

Area = (1/2) * 16 * 8.66
Area ≈ 69.28 in^2

Therefore, the area of the rhombus is approximately 69.28 square inches.

Now, let's move on to the problem of finding the area of a regular decagon with a radius of 4 cm.

A regular decagon is a polygon with 10 congruent sides and 10 congruent angles. To find the area of a regular decagon, you can use the formula:

Area = (5/4) * s^2 * tan(π/10)

where s is the length of one side of the decagon.

In this case, you are given the radius (4 cm), which is also the distance from the center of the decagon to any vertex. To find the length of one side (s), you can use the formula:

s = 2 * r * sin(π/10)

where r is the radius.

Substituting the given radius (4 cm) into the formula, you get:

s = 2 * 4 * sin(π/10)
s ≈ 5.048 cm

Now that you have the length of one side (s ≈ 5.048 cm), you can plug it into the area formula:

Area = (5/4) * (5.048 cm)^2 * tan(π/10)
Area ≈ 31.91 cm^2

Therefore, the area of the regular decagon is approximately 31.91 square centimeters.

2. Decagon: 10 sides.

Area = 2.94*r^2 = 2.94*4^2 = 47 cm^2.

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