Did it in my head. The diameter of the circle is the hypotenuse of the triangle, with sides of 7 and 12. "a squared + b squared = c squared". So, 7 squared (49) + 12 squared (144) equals 193. Since 14 squared is 196, the hypotenuse length is almost 14, making the radius almost 7. The only given answer that fits is (C).
@@richardhole8429 Yeah! Isn't approximation marvelous for going straight to the heart of many problems? You first ask "how close do I have to get?" The answer? Close enough to eliminate 3 of the given answers. This one is straightforward. But for anyone not completely sure at first, look at the one leg that's length 12. The circle's diameter (hypotenuse) has to be bigger than that, so the radius has to be more than 6. Two answers eliminated immediately. Then hold one end of the length-7 leg, and use the end that's at the right angle to start drawing a circle. When you get to the diameter of the original circle, look how far it comes along that diameter. You can visually see you're very close to the center. Bingo! Radius is about 7! You don't even need to calculate at all!
The triangle is right angled, based on the fact that any angle that touches the circumference and the end points of the diameter is 90°. 12²+7²=144+49=193 √193 is just under 14, meaning when you divide it by 2, it will be just under 7. Therefore, it is 6.94
I solved this problem is a different way. I joined the upper vertex to the centre of the circle and that new line resulted in two isosceles triangles with the circle's radius being the common equal-lengths of two of the the sides of each triangle. If I called the common angles of the left-hand triangle as A and those of the right triangle as B, then the original triangle's upper vertex is equal to A + B and in that large triangle 2A + 2B = 180°. So A + B = 90°; i.e. the upper vertex is a right angle. I hope I've explained it sufficiently well and I realise diagrams would make it much more obvious.
I forgot to complete the problem. The hypotenuse is therefore equal to the square root of 12² + 7², or 144 + 49; i,e, the square root of 193 and that works out to be just under 13.9. So the radius is just under 6.95
Useful info. The degrees in an inscribed angle on a circle equal half the number in the arc it sweeps out. Another concept I never soaked up till tonight. Learn something every day on utube (if I visit the right videos).
Regarding decimal places, I was taught that if the final answer needed to be given to x places, then all interim calculations should be done to at least x+1 places, with only the final result being taken to x places. That way there's no chance of multiple round-offs creating a cumulative error. Although luckily in this case it wouldn't have mattered, if only the rounding had been done properly. Which it wasn't. 6.945 rounds off (for anyone but a banker) to 6.95, not 6.94.
@dazartingstall: Unfortunately your statements are - partly - wrong as well. But first of all the item which you explained (almost) correctly: When the requested number of decimals is x, you should calculate interim results by AT LEAST x+1 decimals. Now the remaining errors: 1. You conclude: "That way there's no chance of multiple round-offs creating a cumulative error." That's wrong. Cumulative errors can not be avoided. They are not so frequent as correct results, but that happens coincidentially. 2. 6.945 does not round of to 6.95. The scientific rounding rule is: 5 is rounded to an even decimal in the place left to it, except there are further decimal unequal 0 right from it. 3. The bankers rule is right the opposite: So you obviously mixed them up. In this case, having reached the interim result of 13.89 for the sqare root of 193 has to bee handled according to rule 2., because 193 has no rational square root and therefore the 13.89 is followed by an infinite number of decimals after the 2nd digit. So we know, that 13.89 / 2 > 6.945 and we have - banker or scientist is irrelevant - to round to 6.95.
Just in my head, I assumed the triangle was a right triangle (not remembering that it HAD to be). I applied Pythagorus' Theorem and quickly came up with the answer. However, simply comparing the short side to the given choices, the answer becomes apparent visually without resorting to any math at all. To make the problem a little more difficult, the given choices should have been just tenths of a unit different, forcing the solver to actually do the math to home-in on the correct answer. Sometimes a problem is much simpler than it appears at first glance. I got it very quickly, especially for a 73 year old who was never much good at math. I DO remember certain things from 6th grade.
My answer was 6.95 to 2dp. From 13.892, as ✓193, without the aid of a calculator. You didn't show how you got your a without using a calculator. Would love to see if you used the same method as me. I'm almost 77 years old and am loving your thoroughness in making algebra even more addictive than it already was to me! Thank you. Jeni Bourne 😅
Geometry was my worst math subject in that I went to a new school at that point and missed out on learning well the teaching of working these. Thanks for going over this.
The radius is half the diameter and the diameter is the hypotenuse of the inscribed right triangle. One can find it by using the Pythagoras theorem (12^2 + 7^2 = the hypotenuse / diameter ^2.
Greetings. The answer is 6.94 units. The hypotenuse of the inscribed triangle is actually the diameter of the circle. Therefore, using Pythagoras' theorem we have 12^2+7^2= the diameter squared, and 144+49= diameter, D^2. Moving forward D^2= 193 units. That is D=193^1/2 units, and D=13.89 units. Now, we were asked to determine the radius. Therefore,all that is left to be done is for us to do is to use R=D/2 to get Radius, R=13.89/2 units. R=6.94 units.
A triangle is inscribed in a circle and one of it's sides is the diameter of the circle, so, that's a rectangle triangle and the angle between the side with size 12 and the side of size 7 is 90°. Apply the Pythagoras theorem, we get the diameter. Divide by 2 to get the radius. Answer is C. You can also divide the size of the triangle by 2, making it's sides 6, 3.5 and radius of the circle.
c) 6.94 hypotenuse=2r Inscribed angle is half of angle at same two points of circle, which is, at this picture, 180°= π rad(ians). It means that inscribed angle has measure of 90°=π/2, so, first we have theorem of inscribed angles, and second, we of course have Pytagorean theorem. After that, we just must calculate...
This is the problem with multiple choice testing in mathematics. It is possible to look at this diagram and use simple deduction to figure out the diameter has to be bigger than a) or b), and that the diameter can’t be over 17. So they chooses option c) on a math placement exam and end up quickly falling behind the class because it’s assumed they know how they got the answers. If students were made to write out their maths and answers, then it would be apparent where they need assistance.
Did you just kinda explain the Thalus theorem? I don’t think it amounts to a proof, but kinda getting to it, especially, as long as kids can make sense of it. For these type of geometry problems, I’d love if you would give some historic background. I got into math because of part of the background info my teacher gave. E.g. Appolonius, ‘don’t disturb my circles ‘ really captured me . Don’t know why, don’t know how, but it did 🤓
Thank you Mr. Math RUclips Man (MM-YTM), Sir I just turned the triangle, or rotated it left, you know, in my head, making the 12 the width, or base, the side 7 as the height, then applied Pythagorean theorem. ()12^2) + (7^2) = 144 + 49 = 193 sqrt 2 = ~13.892 / 2 for radius = 6.946 = 6.95
c. Using the Pythagorean Theorem is easy. The trick is knowing that two lines drawn from opposite ends of the diameter to a common point form a right angle.
Used the Pythagorian formula for a right triangle, after checking it was indeed a right triangle, then divided the hypotenuse by 2. Which was 6.94622 the answer 6.94 is correct, but if we are rounding up the third decimal place...should actually be 6.95, but I'm splitting hairs. This problem took me 30 seconds to solve.
Im basically 7th grade but i remember certain things from 5th grade This looks like a 5th grade questions. The shape area and perimeter We should just respect his ways of teaching, I mean, He have tried. It's his way. I have my own way tho Because there is no explanation which is the longest sides, we can just do it like = a² + b² = c², do a² = 7 and b² = 12 So we were looking the third sides of the triangle, so we need to Caculate : so (7 x 7) + (12 x 12 ) = 49 + 144 = 193. So basically : 193 = c² is 13,69, 13,69 is the diameter and the problem told us to find the radius so we divided by two = 13,69 : 2 = 6,945 so basically c. Is the correct answer If you don't want to count ²√193 basically just search the nearest number, 2√196 the answer is 14 divided by 2 is 7, The nearest number to 7 is 6,94 so its the same answer..
How have you managed to round it to 6.94 and not 6.95? Doesn't matter how many or how few decimal points I use for √193, I can't find a way that it rounds to 6.94.
The squaws of the hippopotamus is equal to the sum of the squaws of the other two hides. which means radius is one half the square root of((12x12)+(7x7)) or one half of the square root of 193 or 6.94622... rounded to two decimal places is 6.95, NOT 6.94.
I remember proving that theorem something like 60 years ago in my Plane Geometry course. I don't remember how I proved in now, but what you did doesn't feel like what I did. Of course, you didn't present a complete proof, more like a justification.
Thanks for the lesson on inscribed angles, i just don't remember this from my school days or probably didn't understand it at that time so erased from my memory 😂
It has to be 6.94. 12 squared is 144. 7 squared is 49. So the diameter is the square root of 193. 14 squared is 196. So the radius is a little under half of 14.
I had a terrible geometry teacher and i took BASIC geometry even though it was supposed to be for remedial math students. Still can't find the circumference of a circle 30 years later. But my guess is C
We don’t need that complicated approach to find the angle above the diameter. All triangles circumscribed by a semicircle are right-angled. Thales‘ theorem.
In any triangle where one side is the diameter, with the other two sides meeting at the circumference, the corner created by those two sides will be a right angle ...which he explains.
Moving the point between 12 and 7 to the top of the circle means 12 will decrease and 7 will increase. Since there is this stupid multiple choice I will not calculate but estimate the answer: (12+7) / 2 = 9.5 then Pythagoras 2r² = 9.5² So r is about 6.72... which is close enough with answer C.
I really appreciate the math problem but you are the type of the teacher that talks so much that makes math difficult to understand! Please cut the excessive talking! 3 minutes into this clip I realized that you are trying to prolong this to benefit from the RUclips pay as opposed to teaching something to people! Not interested anymore!!!
The solution a total Yap-fest. You can work out the sum of the squares in you noggin. The diameter on 2 is the answer. I used to get these kind of yappy answers at tech college when a short estimatin eqn would do the trick. Yap Yap ..YAaaaaaaP ... and ..Yap Yap ..YAaaaaaaP ... ... oh byt the way Yap Yap ..YAaaaaaaP ... oh yes I left out Yap Yap ..YAaaaaaaP ... ZZzzzZzZz HuH...ZZzZzzZz... wassat...
Did it in my head. The diameter of the circle is the hypotenuse of the triangle, with sides of 7 and 12. "a squared + b squared = c squared". So, 7 squared (49) + 12 squared (144) equals 193. Since 14 squared is 196, the hypotenuse length is almost 14, making the radius almost 7. The only given answer that fits is (C).
Yeah this was really easy
I did exactly the same.
@@richardhole8429 Yeah! Isn't approximation marvelous for going straight to the heart of many problems? You first ask "how close do I have to get?" The answer? Close enough to eliminate 3 of the given answers. This one is straightforward. But for anyone not completely sure at first, look at the one leg that's length 12. The circle's diameter (hypotenuse) has to be bigger than that, so the radius has to be more than 6. Two answers eliminated immediately. Then hold one end of the length-7 leg, and use the end that's at the right angle to start drawing a circle. When you get to the diameter of the original circle, look how far it comes along that diameter. You can visually see you're very close to the center. Bingo! Radius is about 7! You don't even need to calculate at all!
And here we have two pragmatic ways to solve this problem. I used Pythagorean theorem.
@@martinwalker9386 Yes, and if you're not doing a multiple guess test question or need extended precision, that's the ticket. They're all very useful.
The triangle is right angled, based on the fact that any angle that touches the circumference and the end points of the diameter is 90°.
12²+7²=144+49=193
√193 is just under 14, meaning when you divide it by 2, it will be just under 7. Therefore, it is 6.94
except that it is 6.95
@@lukeknowles5700 look at the options. Where is the option 6.95?
@@Hi-rw8vr Waver-- I was just suggesting what the correct option should have been, had rounding been done properly. You should chill out.
I solved this problem is a different way. I joined the upper vertex to the centre of the circle and that new line resulted in two isosceles triangles with the circle's radius being the common equal-lengths of two of the the sides of each triangle. If I called the common angles of the left-hand triangle as A and those of the right triangle as B, then the original triangle's upper vertex is equal to A + B and in that large triangle 2A + 2B = 180°. So A + B = 90°; i.e. the upper vertex is a right angle. I hope I've explained it sufficiently well and I realise diagrams would make it much more obvious.
I forgot to complete the problem. The hypotenuse is therefore equal to the square root of 12² + 7², or 144 + 49; i,e, the square root of 193 and that works out to be just under 13.9. So the radius is just under 6.95
Useful info. The degrees in an inscribed angle on a circle equal half the number in the arc it sweeps out. Another concept I never soaked up till tonight.
Learn something every day on utube (if I visit the right videos).
Regarding decimal places, I was taught that if the final answer needed to be given to x places, then all interim calculations should be done to at least x+1 places, with only the final result being taken to x places. That way there's no chance of multiple round-offs creating a cumulative error. Although luckily in this case it wouldn't have mattered, if only the rounding had been done properly. Which it wasn't. 6.945 rounds off (for anyone but a banker) to 6.95, not 6.94.
My bank pays interest owed to me using John's method.
Any money owed by me is calculated using the correct method (explained by you).
Funny that!
@dazartingstall: Unfortunately your statements are - partly - wrong as well. But first of all the item which you explained (almost) correctly: When the requested number of decimals is x, you should calculate interim results by AT LEAST x+1 decimals. Now the remaining errors:
1. You conclude: "That way there's no chance of multiple round-offs creating a cumulative error." That's wrong. Cumulative errors can not be avoided. They are not so frequent as correct results, but that happens coincidentially.
2. 6.945 does not round of to 6.95. The scientific rounding rule is: 5 is rounded to an even decimal in the place left to it, except there are further decimal unequal 0 right from it.
3. The bankers rule is right the opposite: So you obviously mixed them up.
In this case, having reached the interim result of 13.89 for the sqare root of 193 has to bee handled according to rule 2., because 193 has no rational square root and therefore the 13.89 is followed by an infinite number of decimals after the 2nd digit. So we know, that 13.89 / 2 > 6.945 and we have - banker or scientist is irrelevant - to round to 6.95.
Here's a reasonable question: What are the least amount of words needed to explain this?
Just in my head, I assumed the triangle was a right triangle (not remembering that it HAD to be). I applied Pythagorus' Theorem and quickly came up with the answer. However, simply comparing the short side to the given choices, the answer becomes apparent visually without resorting to any math at all. To make the problem a little more difficult, the given choices should have been just tenths of a unit different, forcing the solver to actually do the math to home-in on the correct answer. Sometimes a problem is much simpler than it appears at first glance. I got it very quickly, especially for a 73 year old who was never much good at math. I DO remember certain things from 6th grade.
My answer was 6.95 to 2dp. From 13.892, as ✓193, without the aid of a calculator. You didn't show how you got your a without using a calculator. Would love to see if you used the same method as me. I'm almost 77 years old and am loving your thoroughness in making algebra even more addictive than it already was to me! Thank you. Jeni Bourne 😅
Geometry was my worst math subject in that I went to a new school at that point and missed out on learning well the teaching of working these. Thanks for going over this.
The radius is half the diameter and the diameter is the hypotenuse of the inscribed right triangle. One can find it by using the Pythagoras theorem (12^2 + 7^2 = the hypotenuse / diameter ^2.
Greetings. The answer is 6.94 units. The hypotenuse of the inscribed triangle is actually the diameter of the circle. Therefore, using Pythagoras' theorem we have
12^2+7^2= the diameter squared,
and 144+49= diameter, D^2. Moving forward D^2= 193 units. That is
D=193^1/2 units, and D=13.89 units.
Now, we were asked to determine the radius. Therefore,all that is left to be done is for us to do is to use
R=D/2 to get Radius, R=13.89/2 units. R=6.94 units.
d. 8.61
C 6.94
A triangle is inscribed in a circle and one of it's sides is the diameter of the circle, so, that's a rectangle triangle and the angle between the side with size 12 and the side of size 7 is 90°. Apply the Pythagoras theorem, we get the diameter. Divide by 2 to get the radius. Answer is C.
You can also divide the size of the triangle by 2, making it's sides 6, 3.5 and radius of the circle.
c) 6.94
hypotenuse=2r
Inscribed angle is half of angle at same two points of circle, which is, at this picture, 180°= π rad(ians).
It means that inscribed angle has measure of 90°=π/2, so, first we have theorem of inscribed angles, and second, we of course have Pytagorean theorem. After that, we just must calculate...
Thales theorem.
Yes: All triangles circumscribed by a semicircle are right-angled.
This is the problem with multiple choice testing in mathematics. It is possible to look at this diagram and use simple deduction to figure out the diameter has to be bigger than a) or b), and that the diameter can’t be over 17. So they chooses option c) on a math placement exam and end up quickly falling behind the class because it’s assumed they know how they got the answers. If students were made to write out their maths and answers, then it would be apparent where they need assistance.
Did you just kinda explain the Thalus theorem? I don’t think it amounts to a proof, but kinda getting to it, especially, as long as kids can make sense of it. For these type of geometry problems, I’d love if you would give some historic background. I got into math because of part of the background info my teacher gave. E.g. Appolonius, ‘don’t disturb my circles ‘ really captured me . Don’t know why, don’t know how, but it did 🤓
Thank you Mr. Math RUclips Man (MM-YTM), Sir I just turned the triangle, or rotated it left, you know, in my head, making the 12 the width, or base, the side 7 as the height, then applied Pythagorean theorem. ()12^2) + (7^2) = 144 + 49 = 193 sqrt 2 = ~13.892 / 2 for radius = 6.946 = 6.95
c. Using the Pythagorean Theorem is easy. The trick is knowing that two lines drawn from opposite ends of the diameter to a common point form a right angle.
Thank you.
👑 EXACTLY = 6,95. So option C is closest! 👑
Only when rounded to 2 decimal places of course
Used the Pythagorian formula for a right triangle, after checking it was indeed a right triangle, then divided the hypotenuse by 2. Which was 6.94622 the answer 6.94 is correct, but if we are rounding up the third decimal place...should actually be 6.95, but I'm splitting hairs. This problem took me 30 seconds to solve.
Since the answer calculates out to 6.946, the two decimal place answer should be 6.95 instead of truncating the 6.
thank you
Im basically 7th grade but i remember certain things from 5th grade
This looks like a 5th grade questions. The shape area and perimeter
We should just respect his ways of teaching, I mean, He have tried. It's his way. I have my own way tho
Because there is no explanation which is the longest sides, we can just do it like = a² + b² = c², do a² = 7 and b² = 12
So we were looking the third sides of the triangle, so we need to Caculate : so (7 x 7) + (12 x 12 ) = 49 + 144 = 193.
So basically : 193 = c² is 13,69, 13,69 is the diameter and the problem told us to find the radius so we divided by two = 13,69 : 2 = 6,945 so basically c. Is the correct answer
If you don't want to count ²√193 basically just search the nearest number, 2√196 the answer is 14 divided by 2 is 7, The nearest number to 7 is 6,94 so its the same answer..
it is the square roor of 193 divided y 2, so the result is 6.94. Therefore the answer is c)
got it C pyth for the hyp and /2 = radius easy thanks for the fun
How have you managed to round it to 6.94 and not 6.95?
Doesn't matter how many or how few decimal points I use for √193, I can't find a way that it rounds to 6.94.
The squaws of the hippopotamus is equal to the sum of the squaws of the other two hides. which means radius is one half the square root of((12x12)+(7x7)) or one half of the square root of 193 or 6.94622... rounded to two decimal places is 6.95, NOT 6.94.
I remember proving that theorem something like 60 years ago in my Plane Geometry course. I don't remember how I proved in now, but what you did doesn't feel like what I did. Of course, you didn't present a complete proof, more like a justification.
How did you spin this out to 17 min when it takes less than a minute to solve?
Thanks for the lesson on inscribed angles, i just don't remember this from my school days or probably didn't understand it at that time so erased from my memory 😂
Solution starts at 7:45. Math does't start until 14:00.
Rounding error, correct answer is 6.95 with rounding to 2 decimal places.
Diameter is the square root of 193, which is close to 14. Radius is 1/2 of diameter, so the only reasonable answer is c.
It has to be 6.94. 12 squared is 144. 7 squared is 49. So the diameter is the square root of 193. 14 squared is 196. So the radius is a little under half of 14.
I had a terrible geometry teacher and i took BASIC geometry even though it was supposed to be for remedial math students. Still can't find the circumference of a circle 30 years later. But my guess is C
The CORRECT answer is 6.95 units (rounded to two places to the right of the decimal point).
Far too long-winded. You need less talk more maths
c) 6.94
Actually, it is closer to 6.95 than 6.94.
6.94
8.61
C 6.94
You always find a complicated and confusing method .
The answer correctly rounded to the second decimal place is 6.95, so none of listed choices are correct.
We don’t need that complicated approach to find the angle above the diameter. All triangles circumscribed by a semicircle are right-angled. Thales‘ theorem.
This video is unnecessarily long and can confuse junior class students.
r=6.94
6,94
only true if triangle is a right triangle ...which u did not specify?
In any triangle where one side is the diameter, with the other two sides meeting at the circumference, the corner created by those two sides will be a right angle ...which he explains.
It can only be solved if it is a right triangle.
C..... square root of (144+49)
Far too long. Cut the explanation time 10 fold and I'll stay interested.
Wouldn’t it be 6.95 not 6.94 because 6.945 rounds up?
What if that triangle wasn't a right triangle?
The squaw on the hippopotamus hide is equal to the sum of the squaws on the other two hides. Solution to indian bride selection problem.
Moving the point between 12 and 7 to the top of the circle means 12 will decrease and 7 will increase. Since there is this stupid multiple choice I will not calculate but estimate the answer: (12+7) / 2 = 9.5 then Pythagoras 2r² = 9.5²
So r is about 6.72... which is close enough with answer C.
√193/2
Which= c)
C is the only answer that makes sense, best guess.
Thank you Sir, may God bless, but, maybe you've heard this before, yes, when the heck are we going to use this? lol?
Cis the right answer
C)
Yeah, pythagora's theorem solved it in a minute👍👍why angle😁😁😁
C
6.94. no calculator needed for this
Answer: none of them - the correct answer to 3 significant figures is 9.95 cm.
I think it’s C, but that multiplication in your head is a little rough.
B
How to make a short story long! Takes less than 15".
+1
Mental arithmetic in 9 seconds. This video takes more than 15 minutes - rediculous.
I really appreciate the math problem but you are the type of the teacher that talks so much that makes math difficult to understand! Please cut the excessive talking! 3 minutes into this clip I realized that you are trying to prolong this to benefit from the RUclips pay as opposed to teaching something to people! Not interested anymore!!!
That's both rude and wrong.
Christ, Americans talk so much. Short it down to 2 minutes dude.
The solution a total Yap-fest. You can work out the sum of the squares in you noggin. The diameter on 2 is the answer. I used to get these kind of yappy answers at tech college when a short estimatin eqn would do the trick. Yap Yap ..YAaaaaaaP ... and ..Yap Yap ..YAaaaaaaP ... ... oh byt the way Yap Yap ..YAaaaaaaP ... oh yes I left out Yap Yap ..YAaaaaaaP ...
ZZzzzZzZz HuH...ZZzZzzZz... wassat...
You bad teacher to much talk for nothing
6.94
C 6.94
r = 6.94
C
B
6.94
6.94
6.94
6.94
6.94
6.94
C
C
C
C
C