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Show that the following triangles, which have sides of the given lengths_ are each right triangles: a = 2r + 1 b 2r2 2r , and c = 272 2r + [_ where a real number...

Question

Show that the following triangles, which have sides of the given lengths_ are each right triangles: a = 2r + 1 b 2r2 2r , and c = 272 2r + [_ where a real number

Show that the following triangles, which have sides of the given lengths_ are each right triangles: a = 2r + 1 b 2r2 2r , and c = 272 2r + [_ where a real number



Answers

CIRCUMSCRIBED AND INSCRIBED CIRCLES
Let $R$ and $r$ be the radii of the circumscribed and inscribed circles of a triangle $ABC$, respectively (see figure), and let

$s = \dfrac{a+b+c}{2}$.


(a)
Prove that $2R = \dfrac{a}{\sin\ A} = \dfrac{b}{\sin\ B} = \dfrac{c}{\sin\ C}$.


(b)
Prove that $r = \sqrt{\dfrac{(s-a)(s-b)(s-c)}{s}}$.

We are trying to prove that these three points will form a right triangle. The there are two distances that are very easy to find The distance from 00-40. Since there's only a change in the X is just that change in the X. So the length of that leg is four. The distance from 40 to 43 is just the change in the Y coordinate. Since that's the only change. So that's three. The blue side is our hypothesis. And we can find that distance using our distance equation. So we're going to take the square root of the difference between the two x values squared plus the distance between The difference between the two Y Values Square. That will give us the square root of four squared plus three squared. Four square plus three squared is 25. In the square to 25 is five. And the pythagorean theorem does hold true. Five squared is equal to four squared plus three squared. Okay.

We're trying to show that these these three points will form a right triangle. The leg distances are very easy to find because if we look at the horizontal leg, the only changes in the X value. So if it's going from the x point of one to the export of 13, That's the length of 12. The vertical Leg has only a change in the Y values and it's going from 1-6. So that's five for the distance between 11 and 13 6, we will use the distance equation, which is the square root of the difference between the X is squared plus the difference between the wise squared, That's going to give us the square root of 12 squared plus five square, Which is 144 plus 25 on the inside, which would be 169, which is 13. That does indeed make this right triangle. Because the pythagorean theorem would tell us that 13 squared plus 12 square would need to equal five squared. Both sides are 169 and that is true.

We're trying to show that these three points to satisfy the pythagorean theorem to the sides are very easy to find. The red side is a horizontal side and the only change in its coordinates are the change the x coordinates, the distance from negative three X equals negative 32 X equals seven is 10. So the length of that is 10 On the vertical side. We have it going from 712, The only changes in the y coordinates and The difference between four and 1 is three. For the hypotenuse, we were going to use the distance equation, which is the square root of the difference between the X coordinates square plus the difference between the y coordinate square, Well, seven minus negative three. It's really seven plus three. So we have the square root of 10 squared plus three squared. That's going to be 100-plus 9 or the square root Of 109. To show that this satisfies the pythagorean theorem. We will show that the hi pa tennis squared is equal to the sum Of the squares of the two legs. We would end up with a statement 109 equals 100 plus nine or 109 so it's true.

We are trying to show that we are satisfying the Pythagorean theorem and therefore we have a right triangle for the two legs. While the horizontal leg Is going from 2 -2 to 6. Negative too. That is a change only in the Y. The X coordinate And 6 -2 is four. Uh huh. On the vertical line, vertical leg, the only changes in the y coordinate and it's going from Y equals negative 2225 which is a change of seven. For the high pod news, we're going to use the distance equation which is the square root of the difference between the X values squared plus the difference between the y values square. Okay. And the negative two minus five. That's the same thing as negative two, negative five. So we have four square to four squared plus negative seven squared, which would total 16 plus 49 or The Square of 65. And we can show that this is indeed a right triangle. By satisfying the pythagorean theorem and pot news squared equals the sum of the squares of the two legs, Swear to 65 squared is 65, and then we have seven squared plus four square, that's 49 plus 16 or 65.


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