Question
Prove that the angle between the two tangent drawn from an external point to a circle is supplenentary to the angle subtended by the line segments joining the points of contact at the centre.
✓
Answer
Given PA and PB are the tangent drawn from a point P to a circle with centre O. Also, the line segments OA and OB are shown.
To prove: $\angle\text{APB}+\angle\text{AOB}=180^\circ$
Proof:
We know that the tangent is perpendicular to the radius through the point of contact.
$\therefore\text{PA}\perp\text{OA}\Rightarrow\angle\text{OAP}=90^\circ$
$\therefore\text{PB}\perp\text{OB}\Rightarrow\angle\text{OBP}=90^\circ$
$\therefore\angle\text{OBP}+\angle\text{OBP}=90^\circ+90^\circ=180^\circ\dots\text{(i)}$
But, we know that tha sum of all the angles of a quadrilateral is 360º.
$\therefore\angle\text{OAP}+\angle\text{APB}+\angle\text{AOB}+\angle\text{OBP}=360^\circ\dots\text{(ii)}$
From (i) and (ii), we get
$\angle\text{APB}+\angle\text{AOB}=180^\circ$
Hence proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A milk container is made of metal sheet in the shape of frustum of a cone whose volume is $10459\frac{3}{7}\text{cm}^3.$ The radii of its lower and upper circular ends are $8\ cm$ and $20\ cm,$ respectively. Find the cost of metal sheet used in making the container at the rate of $₹\ 1.40\ per\ cm^2.$
→If $(2\sin\theta+3\cos\theta)=2,$ and $(3\sin\theta-2\cos\theta)=\pm3.$
→Solve the following quadratic equation:
$\frac{1}{\text{x}-1}-\frac{1}{\text{x}+5}=\frac{6}{7},$ $\text{x}\neq1,\ -5$
→$\frac{1}{\text{x}-1}-\frac{1}{\text{x}+5}=\frac{6}{7},$ $\text{x}\neq1,\ -5$
If 45 is subtracted from twice the greater of two numbers, it result in the other number. If 21 is subtracted from twice the smaller number, it results in the greater number. Find the numbers.
Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:
2x - 3y + 4 = 0, x + 2y - 5 = 0
2x - 3y + 4 = 0, x + 2y - 5 = 0
From the following data, draw the two types of cumulative frequency curves and determine the median:
|
Height (in cm)
|
Frequency
|
|
140-144
|
3
|
|
144-148
|
9
|
|
148-152
|
24
|
|
152-156
|
31
|
|
156-160
|
42
|
|
160-164
|
64
|
|
164-168
|
75
|
|
168-172
|
82
|
|
172-176
|
86
|
|
176-180
|
34
|
Find the roots of the following equation, if they exist, by applying the quadratic formula:
$3a^2x^2 + 8abx + 4b^2 = 0$, $\text{a}\neq0$
→$3a^2x^2 + 8abx + 4b^2 = 0$, $\text{a}\neq0$
Find two numbers such that the sum of thrice the first and the second is 142, and four times the first exceeds the second by 138.
Three equal circles, each of radius 6cm, touch one another as shown in the figune. find the area enclosed between them. $\big[\text{Take }\pi=3.14\text{ and }\sqrt{3}=1.732.\big]$
→Solve for x and y:
$\frac{35}{\text{x}+\text{y}}+\frac{14}{\text{x}-\text{y}}=19,$ $\frac{14}{\text{x}+\text{y}}+\frac{35}{\text{x}-\text{y}}=37$
→$\frac{35}{\text{x}+\text{y}}+\frac{14}{\text{x}-\text{y}}=19,$ $\frac{14}{\text{x}+\text{y}}+\frac{35}{\text{x}-\text{y}}=37$