A rectangular triangle with 12 cm and 16 cm legs is inscribed in a circle, find the length of the median drawn to the hypotenuse.

| February 6, 2021 | education

A rectangular triangle with 12 cm and 16 cm legs is inscribed in a circle, find the length of the median drawn to the hypotenuse. tangents ma and mv are drawn from point m to a circle with center o, A and B are points of tangency, angle AMO = 40 degrees, find the angles of the triangle MВO

Since a right-angled triangle is inscribed in a circle, the inscribed angle ABC = 900 and rests on the arc AC, then the arcs AC = 90 * 2 = 180, therefore, the hypotenuse AC is the diameter of the circle.

By the Pythagorean theorem, AC^2 = AB^2 + BC^2 = 144 + 256 = 40.

AC = 20 cm.

Then ОВ = ОА = OC = R = OC / 2 = 20/2 = 10 cm.

Answer: The median length is 10 cm.

Figure: 2

Let’s construct the radii ОА and ОВ to the points of tangency. The radius drawn to the tangent point is perpendicular to the tangent, then the triangles AOM and ВОМ are rectangular.

In right-angled triangles AOM and ВOM, the hypotenuse OM is common, OA = OB = R, then the triangles are equal in leg and hypotenuse, which means the angle OMВ = OMA = 40.

Then the ВOМ angle = (90 – 40) = 50.

Answer: The angles of the MВO triangle are 40, 50, 90.



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