Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.1 | Page 83
Four alternative answers for the following question is given. Choose the correct alternative.
Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.
- Only one
- Only two
- Only three
- All
Let P be any point on the arc XZ.
XZ is the diameter of the circle.
∴ ∠XPZ = 90º (Angle in a semi-circle is 90º)
So, ∠XYZ cannot be a right angle.
In ∆YPZ,
∠XYZ > ∠YPZ (An exterior angle of a triangle is greater than its remote interior angle)
⇒ ∠XYZ > 90º (∠YPZ = ∠XPZ)
So, ∠XYZ is an obtuse angle. Therefore, it is not possible that ∠XYZ is an acute angle.
Thus, three of the following statements are true.
Hence, the correct answer is Only three .
Explanation:-
Consider a circle with diameter XZ and let P be any point on the arc XZ.
Since XZ is the diameter of the circle, angle XPZ is a right angle (90°), since it is an angle in a semi-circle.
Suppose angle XYZ is a right angle. Then angle YPZ would be the complement of angle XPZ, which is also a right angle. But this would mean that triangle YPZ is a right triangle, which is not possible since P lies on the arc XZ. Therefore, angle XYZ cannot be a right angle.
In triangle YPZ, angle XYZ is an exterior angle, and it is greater than angle YPZ (which is equal to angle XPZ). This is because an exterior angle of a triangle is always greater than its remote interior angle.
Therefore, we have angle XYZ > angle YPZ = angle XPZ = 90°. So, angle XYZ is an obtuse angle. It is not possible for angle XYZ to be an acute angle since an acute angle is always less than 90°.
Thus, the statement “Only three of the following statements are true” must be true.
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.10 | Page 83
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