Question

Please solve it. ...​

Answer 1

Step-by-step explanation:

All Circles are Similar

The first interesting property of circles that we're going to define is that all circles are similar. In geometry, we say two shapes are similar if we can take one shape and somehow move it and then dilate it in order to match it up completely with the other.

Take for example these two circles.

Two circles

We can move the smaller circle on the right so that its center coincides with the center of the bigger circle.

Concentric circles

We can then enlarge the smaller circle until its radius is exactly the same as that of the larger circle.

Similar circles

Now the two circles match up completely. Therefore, the original two circles are similar. Notice there is nothing special about the original two circles used. The length of each circle's radius is not even defined. This means that we replicate this process with any two circles, leading us to the conclusion that all circles are similar.

Inscribed Angles

When talking about properties of circles, we like to talk about inscribed angles. An inscribed angle is an angle determined by two chords which intersect at the same point on the edge of a circle, like in the picture below.

Inscribed angle

The next property we're going to prove is that inscribed angles with endpoints that intersect the ends of a diameter, as in the picture below, are right angles. Observe that in the picture below, D is the center of the circle and the line segment BC is a diameter. In the following explanation, the labels in the below diagram will be used to help illustrate this proof.

inscribed angle on a diameter

Our end goal is to show that ∠BAC is a right angle, or that m∠BAC = 90°. To help us with this proof, we're going to draw in a line segment from A to D.

null

First, notice that since ΔABC is a triangle, its interior angles add up to 180°. Thus,

m∠BAC + m∠ABC + m∠ACB = 180°.

With the new line segment we drew between A and D, we split ∠BAC into two pieces, ∠BAD and ∠DAC. Since those two angles combine to give us ∠BAC, we know that m∠BAD + m∠DAC = m∠BAC. Thus, by substitution, we have,

m∠BAD + m∠DAC + m∠ABC + m∠ACB = 180°.

This is where the line we drew in becomes especially useful. Take a look at ΔADB and ΔADC. As we look closely at them, we realize they're actually both isosceles triangles! This is because line segments AD, BD, and CD are all radii of our circle meaning that they are all the same length! Since AD = BD, the triangle ΔADB is isosceles, meaning m∠ABC = m∠BAD. Similarly, since AD = CD, ΔADC is isoceles and m∠ACB = m∠DAC.

With these new discoveries in hand, we can take,

m∠BAD + m∠DAC + m∠ABC + m∠ACB = 180°

and use substitution to rewrite it as,

m∠BAD + m∠DAC + m∠BAD + m∠DAC = 180°.

This means that,

2(m∠BAD + m∠DAC) = 180 °,

meaning that,

m∠BAD + m∠DAC = 90°.

Since m∠BAD + m∠DAC = m∠BAC, that means that m∠BAC = 90°, or a right angle!

Tangents and Radii

For our final proof, we're going to try to find another right angle within a circle. This one will be between a radius and a tangent. A tangent is a line that intersects a circle once, and only once, as in the picture below

Answer 2

$\huge \star \underline \bold \red{answer}$

As you know that radius is the half of the circle.

$\small \star \: so \: draw \: \: two \: \: radi$

$and \: then \: two \: angles \: subtended \:$

$what \: we \: gave \: the \: value \: of \small \star2a \: and \: 2b$

$hence \: 2a + 2b = {360}^{0}$

a+b=180degrees.

$\huge \fcolorbox{yellow}{red}{proven}$