To solve the problem of finding the number of sides of a regular polygon given the difference between the interior and exterior angles is 100°, we can follow these steps:
- Understand the relationships between the angles of a polygon:
- The sum of the interior angles of a polygon with \(n\) sides is given by \(180(n - 2)\) degrees.
- The measure of each interior angle of a regular polygon is \(\frac{180(n-2)}{n}\) degrees.
- The exterior angle of a regular polygon is always: \(\frac{360}{n}\) degrees.
- Set up the equation based on the given condition: The difference between the interior and exterior angles is 100°, so:
\frac{180(n-2)}{n} - \frac{360}{n} = 100
- Simplify the equation:
\frac{180n - 360 - 360}{n} = 100
\frac{180n - 720}{n} = 100
- Multiply through by \(n\) to eliminate the fraction:
180n - 720 = 100n
- Simplify and solve for \(n\):
- Rearrange: \(180n - 100n = 720\)
- Simplify: \(80n = 720\)
- Divide by 80: \(n = \frac{720}{80}\)
- Thus, \(n = 9\)
- Conclusion: The polygon must have 9 sides. Therefore, the answer is 9.
This completes our solution. The correct answer is 9 sides, which matches the given correct option.
Similar Questions
- The number of triangles formed by diagonals from one vertex of an octagon is:
- The sum of all interior angles of a 20-sided polygon is:
- If the measure of an exterior angle of a regular polygon is $20^\circ$, determine the total number of its sides.
- What is the minimum distance between centers of two circles having radii 5 cm and 3 cm such that exactly three common tangents exist?
- In a circle, angle subtended by chord AB at the center is $80^\circ$. What is the measure of angle $\angle ACB$, where C lies on the circle?
- In a circle with center O, two chords, AB and CD, cross each other at right angles. if the distance from the center O to chord AB is 3 cm, while the distance from O to chord CD is 4 cm. What is the radius of the circle?
- If the angle subtended by a chord at the center is $150^\circ$, what is the angle subtended at a point on the circle in the same segment?
- A triangle ABC has a circle inscribed within it with sides a, b, c. The area of the triangle is A. What is the radius of the inscribed circle?
- In a triangle ABC, if the angle bisectors of $\angle B$ and $\angle C$ intersect at point I, and $\angle BIC = 110^\circ$, what is the measure of $\angle A$?
- In triangle PQR, ST is parallel to QR. If PS=5cm, SQ=10cm, and PT=6cm, find TR.
Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
