How do we know this line is the median

[egal]

I attached the question and its solution. In the solution thr book says we draw the median AD, but I don't understand how's that line median. I see that it creates the ADC isosceles triangle

 

[Dr.Peterson]

I attached the question and its solution. In the solution the book says we draw the median AD, but I don't understand how's that line median. I see that it creates the ADC isosceles triangle

The question is not "how do we know this line is the median"; it constructed as the median. The question is, how do they conclude that its length is half the hypotenuse?

Since they just say this without explanation, I suspect they have previously proved it as a theorem. But here is one way to see it (in an accurate drawing that has actual right angles):

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The midpoint of the hypotenuse of a right triangle is the circumcenter of the triangle (the center of the circle passing through the vertices). Do you see how to prove this, and how it implies the fact they state?

 

[egal]

View attachment 29183
The question is not "how do we know this line is the median"; it constructed as the median. The question is, how do they conclude that its length is half the hypotenuse?

Since they just say this without explanation, I suspect they have previously proved it as a theorem. But here is one way to see it (in an accurate drawing that has actual right angles):

View attachment 29182​

The midpoint of the hypotenuse of a right triangle is the circumcenter of the triangle (the center of the circle passing through the vertices). Do you see how to prove this, and how it implies the fact they state?

They remarked "we know that, in a right triangle, the median equals half the measure of hypothenuse." I know this is a theorem and I also know the proof of it, this isn't a problem.

The thing I don't understand is that they drew the line from A to a point they named D, and the angle DAC equalled 15 degrees, and for its reason they said it's because AD is median. Because the median has to have the same measure with IDCI, the triangle ADC should be isosceles, thus DAC=ACD=15. But how do I know AD is the median and thereby know DAC is 15?

 

[Dr.Peterson]

The thing I don't understand is that they drew the line from A to a point they named D, and the angle DAC equalled 15 degrees, and for its reason they said it's because AD is median. Because the median has to have the same measure with IDCI, the triangle ADC should be isosceles, thus DAC=ACD=15. But how do I know AD is the median and thereby know DAC is 15?

You know AD is the median because they made it that way, by constructing the midpoint of BC, and calling that D. This is what it means when they say, "We draw the median AD". D is not just some random point.

(I think they're wrong in calling the median they drew "|AD|" rather than "AD"; I imagine their notation is such that |AD| means the length of the median. But I don't think that's what's confusing you.)

But I already said this, so I don't know what you don't understand:

The question is not "how do we know this line is the median"; it [was] constructed as the median.

 

[egal]

You know AD is the median because they made it that way, by constructing the midpoint of BC, and calling that D. This is what it means when they say, "We draw the median AD". D is not just some random point.

(I think they're wrong in calling the median they drew "|AD|" rather than "AD"; I imagine their notation is such that |AD| means the length of the median. But I don't think that's what's confusing you.)

But I already said this, so I don't know what you don't understand:

Okay, understood. "It was constructed as the median", that made sense now. Thanks a lot