An oblique triangle is inscribed in a circle. If one side of the triangle having a length of 10 cm and the angle subtended to that side is 20. Determine the area of the circle.
This problem can be solved using sine law.
By drawing the figure, an oblique triangle is inside the circle
then the angle opposite to its side is 20° , then the subtended angle towards the center of the circle is θ=2θ; 2(20°)= 40°
using summation of the interior angle, 180°= 40 ° + x° + x°
x = (180° - 40°)/2 = 70°
the two remaining angles are both 70 °
using sine law:
(10/sin 40°) = (r/sin 70°)
r =14.6190 cm
then using the formula of an area of a circle:
A = πr²
A = π(14.6190)²
A = 671.4080 cm²
10/d=sin(20)
A=(pi(d)2)/4
ans.A=671.4 sq. unit
Oh, I see we're doing geometry today! Time to put on my mathematical clown nose and entertain you with an answer.
To find the area of the circle, we need to solve this puzzle step by step. Let's start by drawing a little doodle. 🤡
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Alright, our triangle is inscribed in a circle, so let me squeeze in a circle around it. Ta-da! 🤡
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Now, we know that one side of our triangle has a length of 10 cm and an angle of 20 degrees. Take a moment to appreciate this fascinating triangle. 😄
Let's name this side "AB" and the center of the circle "O". Now, it's time to unleash some math magic! 🎩
When an angle is subtended by a chord of a circle, it's twice the angle subtended by the same arc of the circle. So, if the angle at point "A" is 20 degrees, the arc AB will be twice that, meaning 40 degrees. Now we're getting somewhere! 🤔
Next, we need to calculate the length of the arc AB. To do that, we need the circumference of the circle. Luckily, we have a trick up our clown sleeves! 🎉
The formula for the circumference of a circle is C = 2πr, where r is the radius. And guess what, our triangle can offer us insight into the radius!
If we extend the side AB to the center O, we create a radius called "r". By drawing another line from O to the midpoint of AB, we form a right-angled triangle with angles of 20 degrees and 90 degrees. 📐
This means we have a 20-degree angle, a right triangle, and the adjacent side (half of AB, so 5 cm). Time for a little trigonometry! 🌽
Using the cosine function, cos(20) = adjacent / hypotenuse, we can calculate the hypotenuse of this awesome triangle, which is equal to the radius we're looking for. Exciting, isn't it? 🙃
Simplifying the equation, we get:
cos(20) = 5 / r
To find the radius, we rearrange the equation to:
r = 5 / cos(20)
Now we have the radius! Woohoo! So, the arc AB is 40 degrees, and the radius is 5 / cos(20). The length of the arc AB can be found using the formula:
Arc Length = (Angle / 360 degrees) * Circumference
Plugging in the values, we get:
Arc AB = (40 / 360) * (2π * (5 / cos(20)))
Now, you can calculate the length of the arc AB, which is one piece in our puzzle. Almost there! 🎉
Finally, to find the area of the circle, we can use the formula:
Area = π * r^2
Plugging in the radius we found earlier, your calculator (or a friendly mathematician) can help you calculate the area and unveil the final answer! 🎈
Keep practicing these geometric puzzles, and remember, math can be a circus of fun! 🤡
To determine the area of the circle, we need to find the length of the radius first.
Since the oblique triangle is inscribed in the circle, by the Inscribed Angle Theorem, the angle opposite the side of length 10 cm is equal to half of the angle subtended by the same side.
So, the angle opposite the side of length 10 cm is 20/2 = 10 degrees.
We can use the Law of Sines to find the length of the other two sides of the triangle. The Law of Sines states that for any triangle:
sin(A)/a = sin(B)/b = sin(C)/c
Let's label the sides of the triangle:
- opposite angle A, we have side a = 10 cm
- opposite angle B, we have side b (unknown)
- opposite angle C, we have side c (unknown)
Using the Law of Sines, we can write the equation as:
sin(A)/a = sin(B)/b
sin(10°)/10 = sin(B)/b
To solve for b, we rearrange the equation:
b = (sin(B) * 10) / sin(10°)
Now, we need to find angle B. Since the triangle is oblique, the sum of the angles is 180°. We already know angles A and C, so we can find angle B:
angle B = 180° - angle A - angle C
angle B = 180° - 10° - 20°
angle B = 150°
Now, we can substitute the values into the equation to find the length of side b:
b = (sin(150°) * 10) / sin(10°)
Using a scientific calculator, we find:
b ≈ 44.0402 cm
By the Law of Cosines, we can find the length of side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
c^2 = 10^2 + 44.0402^2 - 2 * 10 * 44.0402 * cos(20°)
Using a scientific calculator, we find:
c ≈ 45.1557 cm
Now that we have the length of the radius, which is equal to the longest side of the oblique triangle inscribed in the circle, we can calculate the area of the circle.
The equation for the area of a circle is:
Area = π * r^2
Area = π * (c/2)^2
Substituting the value of c, we get:
Area ≈ π * (45.1557/2)^2
Using a calculator, we can find the approximate value of the area of the circle.
if you mean the angle is 20°, then the radius r of the circle can be found using'
5/r = sin10°
Now you know r, so you can find the area of the circle...
The solutions above are wrong.
sin A = opposite / hypotenuse
We know the angle A = 20 and hypotenuse = 10 cm
Therefore:
Sin 20 = O/ 10
O =3.42 cm
Since circle is inscribed in the triangle, the opposite side is the diameter. D = 3.42 cm so it makes R = 1.71 cm
Area of circle is pi * r²
A = 9.18 cm²