If the diagonals of a cyclic quadrilateral are diameters of the circle through the opposite vertices of the quadrilateral, prove that the quadrilateral is a rectangle
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Given,
ABCD is a cyclic quadrilateral and AC and BD are the diameters of the circle.
To prove: ABCD is a rectangle.
Proof: To prove that ABCD is a rectangle, we only need to prove that its all angles are 90 degrees.
In semi-ciecle ADC,
angle ADC = 90 degrees (since, the line segment from the two ends of the diameter of a semi-circle makes 90 degrees on the circumference.)
Similarly, we can prove that angle ABC, angle BCD and angle DAB is equal to 90 degrees.
Therefore, ABCD is a rectangle.
Hence, Proved.
ABCD is a cyclic quadrilateral and AC and BD are the diameters of the circle.
To prove: ABCD is a rectangle.
Proof: To prove that ABCD is a rectangle, we only need to prove that its all angles are 90 degrees.
In semi-ciecle ADC,
angle ADC = 90 degrees (since, the line segment from the two ends of the diameter of a semi-circle makes 90 degrees on the circumference.)
Similarly, we can prove that angle ABC, angle BCD and angle DAB is equal to 90 degrees.
Therefore, ABCD is a rectangle.
Hence, Proved.
sterlinpeter:
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Hi there!
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For solutions, Refer to the attached picture.
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Let's see some related topics :
⚫ Circle : The collection of all the points, which are at a fixed distance from a fixed point in a plane, is called a circle.
⚫ Radius : A line joining the centre to the Circumference of the circle, is called radius of a circle.
⚫ Secant : A line intersecting a circle at any two points, is called secant.
⚫ Diameter : A chord passing through the point of the circle, is called diameter. It is the longest chord.
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_______________________
For solutions, Refer to the attached picture.
Regrets for handwriting _/\_
_______________________
Let's see some related topics :
⚫ Circle : The collection of all the points, which are at a fixed distance from a fixed point in a plane, is called a circle.
⚫ Radius : A line joining the centre to the Circumference of the circle, is called radius of a circle.
⚫ Secant : A line intersecting a circle at any two points, is called secant.
⚫ Diameter : A chord passing through the point of the circle, is called diameter. It is the longest chord.
_______________________
Thanks for the question !
☺️❤️☺️
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