ABC is a triangle inscribed in a circle centre O..angle ACB =40¡ã and line And =xcm.calculate the radius of the circle.
To calculate the radius of the circle inscribed in triangle ABC, we can use the properties of an inscribed triangle.
1. First, draw a diagram of triangle ABC inscribed in the circle with center O.
- Label the angles: angle ACB = 40 degrees.
- Label the sides: let the length of side AC = a cm and side BC = b cm.
2. Since triangle ABC is inscribed in a circle, we can use the property that the angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at the circumference.
- In this case, angle ACB is subtended by arc AB at the center of the circle. So, angle AOB = 2 * angle ACB = 2 * 40 degrees = 80 degrees.
3. Now, let's draw a perpendicular from the center O to the side AB, and let's call it OD.
- Since OD is perpendicular to AB, angle AOD is a right angle, which means we have a right-angled triangle AOD.
4. In right-angled triangle AOD, OD is the perpendicular from the center O to the side AB, and AD is the radius of the circle.
- We need to find the value of AD (radius of the circle).
5. Next, let's analyze the right-angled triangle ABC.
- Angle ACB is 40 degrees, which means angle CAB is 90 degrees - angle ACB = 90 degrees - 40 degrees = 50 degrees.
- Angle BAC is also 50 degrees because the triangle is isosceles (AC = BC).
- Therefore, we have a right-angled triangle ABC with angles 90 degrees, 50 degrees, and 40 degrees.
6. Now, let's focus on triangle ABC and side AC.
- In triangle ABC, angle BAC is 50 degrees and angle AOB (subtended by arc AB) is 80 degrees.
- Since angle BAC is half of angle AOB (by the property of inscribed angles), we can conclude that triangle ABC is similar to triangle AOB.
7. Since triangles ABC and AOB are similar, the corresponding sides are proportional.
- AC / AO = BC / BO = AB / AB (which simplifies to 1)
- AC / R = BC / R = AB / D (where R is the radius, and D is the length of side AB)
8. We can simplify the above proportions.
- AC / R = BC / R => AC = BC (since the triangle is isosceles)
- AC = BC = a cm (let this be equation 1)
- BC / R = AB / D => BC = AB * R / D (let this be equation 2)
9. We can equate equations 1 and 2.
- a = AB * R / D
10. We also know that AB = 2AD, since AD is the radius of the circle.
- a = 2AD * R / D
11. Solving for the radius R, we can rearrange the equation.
- R = (a * D) / (2 * AD)
12. Now, we need to find the value of AD.
- AD can be found using trigonometry in right-angled triangle AOD.
- In triangle AOD, angle AOD is 80 degrees, so angle OAD is (180 degrees - 80 degrees) / 2 = 50 degrees.
13. In right-angled triangle AOD, we can use trigonometry to find the value of AD.
- tan(angle OAD) = AD / OD
- tan(50 degrees) = AD / (a/2 cm) (since OD = a/2 cm)
- AD = (a/2 cm) * tan(50 degrees)
14. Now that we know the value of AD, we can substitute it back into the equation for R.
- R = (a * D) / (2 * (a/2 cm) * tan(50 degrees))
- R = (a * D) / (a * tan(50 degrees))
- R = D / tan(50 degrees)
15. Finally, substitute the given value of x cm for D (length of side AB) and calculate the radius R.
- R = x cm / tan(50 degrees)
Therefore, the radius of the circle is x cm / tan(50 degrees).
To find the radius of the circle, we can use the following steps:
Step 1: Draw a diagram of the situation. Label the points A, B, C, and O on the circle. Also, draw a line segment from point O to the midpoint of line segment AB, and label this point as D. Let us assume that line segment AD has a length of x cm.
Step 2: Observe that triangle AOD is a right triangle, as the line segment that joins the center of a circle to the midpoint of a chord is perpendicular to the chord.
Step 3: Since triangle AOD is right-angled, we can use the trigonometric relationship between the sides of a right triangle to find the length of OD.
Step 4: In triangle AOD, the angle AOD is equal to twice angle ACB, which means it is equal to 80 degrees.
Step 5: Since we know the angle and the length of AD, we can use the trigonometric function cosine to find the length of OD. The cosine of an angle is equal to the adjacent side divided by the hypotenuse side.
So, cos(80 degrees) = OD / AD.
Step 6: Simplify the equation and solve for OD.
cos(80 degrees) = OD / x.
Step 7: Calculate the value of OD.
OD = x * cos(80 degrees).
Step 8: To find the radius of the circle, we need to find the length of OC, which is equal to OD plus the radius of the circle.
OC = OD + r.
Step 9: Since OC is also the radius of the circle, we can write the equation as:
r = x * cos(80 degrees).
Therefore, the radius of the circle is x * cos(80 degrees) cm.