Draw Concentric Circles in Word
In Geometry, the objects are said to be concentric when they share a common centre. Circles, spheres, regular polyhedra, regular polygons are concentric every bit they share the aforementioned center point. In Euclidean Geometry , two concentric circles should have dissimilar radii from each other. In this article, you will learn what are concentric circles, the theorem on concentric circles, the region betwixt the concentric circles, equations, and examples in detail.
Concentric Circles Meaning
The circles with a common centre are known every bit concentric circles and have different radii. In other words, information technology is defined equally two or more circles that have the same centre point. The region between two concentric circles are of different radii is known equally an annulus.
Concentric Circumvolve Equations
Allow the equation of the circle with centre (-g, -f) and radius √[gtwo+f2-c] be
x2 + y2 + 2gx + 2fy + c =0
Therefore, the equation of the circle concentric with the other circle exist
x2 + y2 + 2gx + 2fy + c' =0
Information technology is observed that both the equations have the same centre (-g, -f), but they take different radii, where c≠ c'
Similarly, a circle with middle (h, k), and the radius is equal to r, and then the equation becomes
( x – h )2 + ( y – k )ii = r2
Therefore, the equation of a circle concentric with the circumvolve is
( x – h )2 + ( y – grand )2 = r1 2
Where r ≠ r1
Past assigning dissimilar values to the radius in the above equation, we shall become a family of circles.
Concentric Circles – Theorem
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
Proof
Given:
Consider two concentric circles C1 and Cii, with centre O and a chord AB of the larger circumvolve C1, touching the smaller circle C2 at the point P as shown in the effigy below.
Construction:
Bring together OP.
To prove: AP = BP
Proof:
Since AB is the chord of larger circle Cane, it becomes the tangent to C2 at P.
OP is the radius of circle Ctwo.
Nosotros know that the radius is perpendicular to the tangent at the signal of contact.
So, OP ⊥ AB
Now AB is a chord of the circle C1 and OP ⊥ AB.
Therefore, OP is the bisector of the chord AB.
Thus, the perpendicular from the centre bisects the chord, i.east., AP = BP.
Region Between Concentric Circles
Every bit mentioned higher up, the region between two concentric circles is called the annulus. However, we can detect the perimeter and surface area of the annulus using appropriate formulas. The area of the annulus is calculated by subtracting the expanse of smaller circles from the area of the larger circle.
Suppose R is the radius of the larger circle and r is the radius of the smaller circle such that the area of the region divisional by these two circles is given by:
Area of annulus = πR2 – πr2
Learn more nearly annulus here.
Concentric Circumvolve Examples
Question: Find the equation of the circle concentric with the circle x2 + y2 + 4x – 8y – half-dozen =0, having the radius double of its radius.
Solution:
Given, circle equation: ten2 + yii + 4x – 8y – 6 =0
Nosotros know that the equation of the circle is ten2 + y2 + 2gx + 2fy + c =0
From the given equation, the centre bespeak is (-2, 4)
Therefore, the radius of the given equation will be
r = √[thou2+f2-c]
r = √[4+16+6]
r = √26
Let R be the radius of the concentric circle.
It is given that, the radius of the concentric circle is double of its radius, then
R = 2r
R = two√26
Therefore, the equation of the concentric circle with the radius R and the middle point (-g, -f ) is
( x – g )2 + ( y – f )2 = R2
(x + 2)2 + ( y – 4 )ii = (2√26 )2
tentwo + 4x + 4 + y2 – 8y + sixteen = 4 (26)
xii + y2 + 4x – 8y +20 = 104
10ii + y2 + 4x – 8y – 84 = 0
Question 2:
Find the surface area betwixt ii concentric circles whose diameters are 35 cm and 21 cm.
Solution:
Given,
The diameters of the two circles are 35 cm and 21 cm.
So, R = 35/2 cm
r = 21/2 cm
Surface area of between concentric circles = πR2 – πrii
= (22/seven) × (35/2) × (35/two) – (22/two) × (21/2) × (21/two)
= (22/7)[(35/2) × (35/ii) – (21/2) × (21/ii)]
= (22/7) × [(352 – 212)/4]
= (22/7) × (56 × fourteen)/4
= 616 cm2
Stay tuned with BYJU'S – The Learning App to learn Maths-related concepts, and also sentinel engaging videos to acquire with ease.
Source: https://byjus.com/maths/concentric-circles/
0 Response to "Draw Concentric Circles in Word"
Postar um comentário