Finding the radius of the circumscribed circle in an isosceles triangle by knowing the hypotenuse and tangent

[Alex56541]

I have a triangle ABC, where the angle of BAC = a (alpha). AB=10 and tangent a = _/7. I have to find the radius of the circumscribed circle.
I started by trying to reverse the tg formula and got that a=_/7 * b which lead me to the answer that R = 5, but then I saw the correct answer and it was wrong. Soon after, I realized that since a=b, that there is no way b can equal _/7 * b. I tried splitting the triangle with altitude, but I'm afraid I can't go further. The correct answer is in the picture.

 

[skeeter]

I have a triangle ABC, where the angle of BAC = a (alpha). AB=10 and tangent a = _/7. I have to find the radius of the circumscribed circle.
I started by trying to reverse the tg formula and got that a=_/7 * b which lead me to the answer that R = 5, but then I saw the correct answer and it was wrong. Soon after, I realized that since a=b, that there is no way b can equal _/7 * b. I tried splitting the triangle with altitude, but I'm afraid I can't go further. The correct answer is in the picture.View attachment 22559

You are given one side, AB = 10, and [math]\tan{\alpha} = \sqrt{7}[/math] with no diagram or other information about the triangle in question?

 

[Alex56541]

You are given one side, AB = 10, and [math]\tan{\alpha} = \sqrt{7}[/math] with no diagram or other information about the triangle in question?

Oh, I am sorry. I thought I wrote it but I guess not. The triangle is an isosceles triangle where AC = BC. But nothing besides that.

 

[skeeter]

Let point A be at the origin and side AB lie on the x-axis.

The circumcenter lies on the altitude from C to AB. Let the point [MATH]D = (5,y)[/MATH] be the circumcenter, equidistant from all three vertices.

point C is at [MATH](5, 5\sqrt{7})[/MATH]
[MATH]CD = 5\sqrt{7}-y[/MATH]
point A is at [MATH](0,0)[/MATH]
[MATH]AD = \sqrt{5^2 + y^2}[/MATH]
set AD = CD, solve for y, then calculate either length AD or CD (should be the same, R)

 

[Dr.Peterson]

I have a triangle ABC, where the angle of BAC = a (alpha). AB=10 and tangent a = _/7. I have to find the radius of the circumscribed circle.
I started by trying to reverse the tg formula and got that a=_/7 * b which lead me to the answer that R = 5, but then I saw the correct answer and it was wrong. Soon after, I realized that since a=b, that there is no way b can equal _/7 * b. I tried splitting the triangle with altitude, but I'm afraid I can't go further.

The triangle is an isosceles triangle where AC = BC. But nothing besides that.

You haven't defined what you mean by "b", or given any details, so I'm not sure of your work.

But are you aware that the center of the circumcircle can be found by intersecting the perpendicular bisectors?



One of many ways to find the radius (CO) would use a pair of similar triangles in this figure, together with the Pythagorean theorem.