If $$ X Y=30, X Z=26 $$, and $$ J Q=8 $$, find the radius of the circumscribed circle of $$ riangle X Y Z $$, The radius of the circumscribed circle of $$ riangle X Y Z $$ is (Simplify your answer.)

Question

If $$ X Y=30, X Z=26 $$, and $$ J Q=8 $$, find the radius of the circumscribed circle of $$ 	riangle X Y Z $$, The radius of the circumscribed circle of $$ 	riangle X Y Z $$ is (Simplify your answer.)

UpStudy AI · Accepted Answer

We first observe that if points $$ J $$ and $$ Q $$ are the midpoints of segments $$ XY $$ and $$ XZ $$ respectively, then the segment $$ JQ $$ is a midsegment of $$\triangle XYZ$$. In any triangle the segment joining the midpoints of two sides is parallel to the third side and its length is half the length of that side. Therefore,
$$
JQ=\frac{1}{2}YZ.
$$
Since $$JQ=8$$, it follows that
$$
YZ=2\cdot 8=16.
$$

Now the side lengths of $$\triangle XYZ$$ are:
$$
XY=30,\quad XZ=26,\quad YZ=16.
$$

To find the radius $$R$$ of the circumscribed circle of the triangle, we use the formula
$$
R=\frac{abc}{4A},
$$
where $$a$$, $$b$$, and $$c$$ are the side lengths and $$A$$ is the area of the triangle.

**Step 1. Compute the area using Heron’s formula.**

The semiperimeter $$s$$ is
$$
s=\frac{30+26+16}{2}=\frac{72}{2}=36.
$$
Then the area $$A$$ is given by
$$
A=\sqrt{s(s-30)(s-26)(s-16)}=\sqrt{36\cdot6\cdot10\cdot20}.
$$
Calculating the product inside the square root:
$$
36\cdot6=216,\quad 216\cdot10=2160,\quad 2160\cdot20=43200.
$$
Thus,
$$
A=\sqrt{43200}.
$$
Notice that
$$
43200=1600\cdot27,
$$
so
$$
A=\sqrt{1600\cdot27}=\sqrt{1600}\sqrt{27}=40\cdot3\sqrt{3}=120\sqrt{3}.
$$

**Step 2. Use the extended Law of Sines formula for the circumradius.**

The product of the side lengths is
$$
30\cdot26\cdot16=12480.
$$
Substitute into the formula for $$R$$:
$$
R=\frac{12480}{4\cdot120\sqrt{3}}.
$$
Simplify the denominator:
$$
4\cdot120\sqrt{3}=480\sqrt{3}.
$$
Thus,
$$
R=\frac{12480}{480\sqrt{3}}.
$$
Dividing numerator and denominator,
$$
\frac{12480}{480}=26,
$$
so
$$
R=\frac{26}{\sqrt{3}}.
$$

We can rationalize the denominator:
$$
R=\frac{26\sqrt{3}}{3}.
$$

Thus, the radius of the circumscribed circle of $$\triangle XYZ$$ is
$$
\boxed{\frac{26\sqrt{3}}{3}}.
$$
The radius of the circumscribed circle of $$ \triangle XYZ $$ is $$ \frac{26\sqrt{3}}{3} $$.

If , and , find the radius of the circumscribed circle of ,
The radius of the circumscribed circle of is
(Simplify your answer.)

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If , and , find the radius of the circumscribed circle of , The radius of the circumscribed circle of is (Simplify your answer.)