The proof that point $(1, \sqrt{3})$ lies on the circle that is centered at the origin and contains the point $(0,2)$ is found in the table below. What is the justification for the 5th statement?
| Statement | Justification |
| --------- | ------------- |
| A circle is centered at $(0,0)$ and contains the point $(0,2)$. | Given |
| The radius of the circle is the distance from $(0,0)$ to $(0,2)$. | Definition of radius |
| The distance from $(0,0)$ to $(0,2)$ is
$\sqrt{(0-0)^2+(2-0)^2}=\sqrt{2^2}=2$ | Distance formula |
| If $(1, \sqrt{3})$ lies on the circle it must be the same distance from the center as $(0,2)$. | Definition of a circle |
| The distance from $(1, \sqrt{3})$ is
$\sqrt{(0-1)^2+(0-\sqrt{3})^2}=\sqrt{1+3}=2$ | |
Since $(1, \sqrt{3})$ is 2 units from $(0,0)$, it lies on a circle that is centered at the origin and contains the point $(0,2)$.
Answer (2)
The problem requires identifying the justification for a statement in a geometric proof.
The statement calculates the distance between two points using the distance formula.
The distance formula is ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Therefore, the justification for the 5th statement is the distance formula. D i s t an ce f or m u l a
Explanation
Problem Analysis The problem provides a proof that the point ( 1 , 3 ) lies on a circle centered at the origin and containing the point ( 0 , 2 ) . The goal is to identify the justification for the 5th statement in the proof, which calculates the distance between the origin ( 0 , 0 ) and the point ( 1 , 3 ) .
Identifying the Justification The 5th statement calculates the distance between the point ( 1 , 3 ) and the origin ( 0 , 0 ) using the formula ( 0 − 1 ) 2 + ( 0 − 3 ) 2 = 1 + 3 = 2 . This calculation is a direct application of the distance formula.
Explanation of the Distance Formula The distance formula is used to calculate the distance between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) in a coordinate plane, and it is given by:
( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
In this case, ( x 1 , y 1 ) = ( 0 , 0 ) and ( x 2 , y 2 ) = ( 1 , 3 ) . Plugging these values into the distance formula, we get:
( 1 − 0 ) 2 + ( 3 − 0 ) 2 = 1 2 + ( 3 ) 2 = 1 + 3 = 4 = 2
Therefore, the justification for the 5th statement is the distance formula.
Final Answer The justification for the 5th statement is the application of the distance formula.
Examples
The distance formula is a fundamental concept in coordinate geometry and has many real-world applications. For example, it can be used in navigation to calculate the distance between two locations on a map, in physics to determine the displacement of an object, or in computer graphics to calculate the distance between two pixels on a screen. Imagine you're planning a road trip and want to know the distance between two cities. You can use their coordinates on a map and apply the distance formula to find the shortest distance between them. If city A is at coordinates (1, 2) and city B is at coordinates (4, 6), the distance between them is ( 4 − 1 ) 2 + ( 6 − 2 ) 2 = 3 2 + 4 2 = 9 + 16 = 25 = 5 units. This helps you estimate travel time and fuel costs.
The justification for the 5th statement is the distance formula, which calculates the distance between point ( 1 , s q r t 3 ) and the origin ( 0 , 0 ) . Using the distance formula, we find that the distance is 2 , which matches the circle's radius. Therefore, ( 1 , s q r t 3 ) lies on the circle defined by the origin and the point ( 0 , 2 ) .
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