A triangle with side lengths 26, 28, and 30 is constructed so that the longest and shortest sides are tangent to a circle. the third side passes through the center of the circle. compute the radius of the circle.
I DON'T KNOW HOW TO DO THIS AT ALL....
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yes that helps. sorry i feel dumb when I ask questions alerdady on here. =/
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no problem, the feature below is a relatively new feature of jishka.
Im new, and I just noticed that....Thanks
No need to worry, I'll guide you through the process step by step. To find the radius of the circle, we can use the properties of a triangle that is tangent to a circle.
Let's start by drawing the triangle.
First, draw a triangle with side lengths of 26, 28, and 30. Label the longest side as "a" and the other two sides as "b" and "c."
Next, draw the circle and locate its center. The shortest and longest sides of the triangle should be tangent to the circle. The third side passes through the center of the circle.
Now, let's use the properties of a tangent line to solve the problem.
When a line is tangent to a circle, it is perpendicular to the radius at the point of tangency. This means that the segment from the center of the circle to the point of tangency is perpendicular to the tangent line.
In our triangle, the segment from the center of the circle to the point of tangency on the shortest side divides the side into two equal segments. Similarly, the segment from the center of the circle to the point of tangency on the longest side divides the side into two equal segments.
Let's label the radius of the circle as "r." We know that the distance from the center of the circle to the point of tangency on the shortest side is also "r." Using the Pythagorean theorem, we can find the length of the other segment of the shortest side:
((c/2)^2) + (r^2) = (b^2)
Now, substitute the given values into the equation:
((30/2)^2) + (r^2) = (26^2)
Simplifying the equation:
(15^2) + (r^2) = (26^2)
225 + (r^2) = 676
Now, solve for r^2:
(r^2) = 676 - 225
(r^2) = 451
Finally, take the square root of both sides to find the radius, r:
r = √451
r ≈ 21.22