Question

Which one of the following statements is true?
a. Only one line can pass through a single point.
b. There are an infinite number of lines which pass through two distinct points.
c. Two distinct lines cannot have more than one point in common.
d. If two circles are equal, then their radii are not equal.

Answer 1

Hint: In order to find out the true statement among the following given statements, we have to check out each and every statement separately by considering examples. We should be applying the geometric rules according to the statement given regarding points, lines and circles.

Complete step-by-step solution:
Now let us check out the given statements. We have our first statement as-
“Only one line can pass through a single point.”
We all know that a plane consists of a number of points. From the first postulate, we all know that we can draw a line from one point to the other point. This can be shown by the following figure.

From the figure, we can see that we can draw a line from \[A\] to\[B\],\[A\] to\[C\],\[A\] to \[D\] and \[A\] to \[E\]. So this proves that many lines can pass through a single point \[A\]. So we can conclude that infinite lines can pass through a single point.
\[\therefore \] The given statement is false.
The second statement is “There are an infinite number of lines which pas through two distinct points “
Let us plot two points on a plane. As we all know an infinite number of lines can pass through a single point, we will be plotting lines accordingly and check out how many lines are going to pass through the distinct points of the same plane.

From the figure, we can conclude that only one line can pass through two distinct points.
\[\therefore \] The given statement is false.
The third statement is “Two distinct lines cannot have more than one point in common.”
In geometry, a line means the totality but not a portion of it. A physical example of a line is not possible. Since a line does not have definite length and extends indefinitely in both the directions. So it cannot be shown on a paper or a plane.
\[\therefore \] The given statement is true.

The fourth statement is “If two circles are equal, then their radii are not equal.”
When two circles are equal, it means that the radii, the diameter are also equal which simply means that the circles are congruent.
\[\therefore \] The given statement is false.

Note: There are three types of geometry. They are- Euclidean, hyperbolic and elliptical. All the straight lines are congruent. Vertical angles are congruent as they measure to the sum of \[{{180}^{\circ }}\]. The sum of measures of the interior angles of a triangle is \[{{180}^{\circ }}\].