a triangle has sides equal to 68 cm, 77 cm and 75 cm, respectively. find the area of the escribed circle tangent to the shortest side of the triangle.
I see you could not be bothered to study the url I gave you earlier. I'll do most of your work for you.
the area of the triangle, found using the method of your choice, is 2310
so, the radius of the excircle is (68+75+77)/2 - 68 = 2310/42 = 55
thus, the circle's area is 3025π
hope that's right; you can double-check my math.
you forgot to write the formula.. lols
Well, well, well, it seems like we have entered the realm of geometry! Do you know why triangles are always so unstable? Because they're always trying to find balance between their three sides. Let's help our triangle find some stability, shall we?
To find the area of the escribed circle tangent to the shortest side of the triangle, we need to first find the length of the shortest side. So, drumroll, please! The shortest side of the triangle is 68 cm.
Now, it's time to introduce a magical concept called Heron's Formula to calculate the area of the triangle. Heron's Formula states that the area of a triangle with sides a, b, and c, and semiperimeter s, is given by the square root of s multiplied by s minus a multiplied by s minus b multiplied by s minus c. Are you following along?
Let's calculate the semiperimeter:
s = (68 + 77 + 75) / 2 = 110
Now, we can calculate the area of the triangle:
A = sqrt(110 * (110 - 68) * (110 - 77) * (110 - 75))
After doing some math, we find that the area of the triangle is approximately 2772.5 square cm.
Now, let's move on to the escribed circle tangent to the shortest side of the triangle. The radius of the escribed circle can be found using the formula r = area of triangle / semiperimeter. Plugging in the values, we get:
r = 2772.5 / 110 ≈ 25.2 cm
Voila! The radius of the escribed circle is approximately 25.2 cm. Now you have the radius, you can find the area of the circle using the formula A = π * r^2.
Remember, if we're not clowning around with math, we should always use pi ≈ 3.14, just to be precise.
To find the area of the escribed circle tangent to the shortest side of the triangle, we need to first find the length of the altitude from the shortest side to the opposite vertex and then calculate the radius of the escribed circle using the semiperimeter of the triangle.
1. Find the semi-perimeter of the triangle.
The semi-perimeter (s) of a triangle is calculated by adding the lengths of all three sides and dividing the sum by 2.
s = (68 + 77 + 75) / 2
s = 220 / 2
s = 110 cm
2. Find the length of the altitude from the shortest side to the opposite vertex.
We can use Heron's formula to calculate the area (A) of the triangle and then use the formula A = 0.5 * base * height to find the altitude length.
a = 68 cm, b = 77 cm, c = 75 cm
Calculate the area using Heron's formula:
A = sqrt(s * (s - a) * (s - b) * (s - c))
= sqrt(110 * (110 - 68) * (110 - 77) * (110 - 75))
= sqrt(110 * 42 * 33 * 35)
= sqrt(1575300)
≈ 1255.181 cm^2
Now we can solve for the altitude:
Area = 0.5 * base * height
1255.181 = 0.5 * 68 * h
h = 1255.181 / 34
h ≈ 36.869 cm
3. Find the radius of the escribed circle.
The radius of the escribed circle (r) can be calculated using the formula:
r = (Area of triangle) / (s - shortest side)
r = 1255.181 / (110 - 68)
r = 1255.181 / 42
r ≈ 29.885 cm
4. Calculate the area of the escribed circle.
The area of a circle (A) can be calculated using the formula:
A = π * (radius)^2
A = π * (29.885)^2
A ≈ 2809.948 cm^2
Therefore, the area of the escribed circle tangent to the shortest side of the triangle is approximately 2809.948 cm^2.
To find the area of the excircle (or the "escribed circle") that is tangent to the shortest side of the triangle, you can follow these steps:
1. Calculate the semiperimeter of the triangle using the given side lengths. The semiperimeter, denoted as "s," is half the sum of the triangle's sides:
s = (a + b + c) / 2
where a, b, and c represent the side lengths of the triangle. In this case, a = 68 cm, b = 77 cm, and c = 75 cm.
Substitute the values to find the semiperimeter, s.
2. Determine the length of the shortest side of the triangle. In this case, it is 68 cm.
3. Use the formula for the radius of the excircle related to the shortest side of the triangle:
r = (s - a) / 2
where r represents the radius of the excircle and a represents the length of the shortest side of the triangle.
Substitute the values from steps 1 and 2 to find the radius, r.
4. Calculate the area of the excircle using the formula:
A = π * r^2
where A represents the area of the excircle, and r represents the radius found in step 3.
Substitute the value of r and calculate the area using the formula.
By following these steps, you will be able to find the area of the excircle tangent to the shortest side of the triangle.