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Problem 698

Prove analytically that any angle inscribed in a semicircle is a right angle.

Expert verified

Given an inscribed angle \(\angle ACB\) in a semicircle with diameter AB and circle center O, we form isosceles triangles \(\triangle OAC \) and \(\triangle OCB \) by connecting O to A, B, and C. Let \(\angle OCA = \angle OAC = x \) and \(\angle OCB = \angle OBC = y \), so \(x + y = 180^\circ\). Using the Inscribed Angle Theorem, we have \(\angle ACB = \frac{1}{2}(\text{arc}(AB)) = \frac{1}{2}( 2x + 2y )\). Substituting the sum, we find \(\angle ACB = 90^\circ\), proving any angle inscribed in a semicircle is a right angle.

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Chapter 40

As seen on the accompanying graph, \(\mathrm{ABCD}\) is a quadrilateral with vertices at \(\mathrm{A}(2,2), \mathrm{B}(5,-2), \mathrm{C}(9,1)\), and \(\mathrm{D}(6,5)\). Prove that the geometric figure is a rhombus.

Chapter 40

Show that the points \(\mathrm{A}(2,-2), \mathrm{B}(-8,4)\), and \(\mathrm{C}(5,3)\) are the vertices of a right triangle and find its area.

Chapter 40

Prove, in the framework of coordinate geometry, that the sum of the squares of the distances from any point in the plane to two opposite vertices of any rectangle is equal to the sum of the squares of its distances from the other two vertices.

Chapter 40

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Chapter 40

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