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The endpoints of $\overline{ AB }$ are $A (-7,-14)$ and $B (5,10)$. Into which ratio will each point divide $\overline{ AB }$ ?
1:2
4:3
2:1
3:1
$C (-3,-6)$
Answer (2)
Calculate the distance A C using the distance formula: A C = (( − 3 ) − ( − 7 ) ) 2 + (( − 6 ) − ( − 14 ) ) 2 = 80 .
Calculate the distance CB using the distance formula: CB = ( 5 − ( − 3 ) ) 2 + ( 10 − ( − 6 ) ) 2 = 320 .
Determine the ratio A C : CB by dividing the distance A C by the distance CB : CB A C = 320 80 = 2 1 .
The ratio in which point C divides the line segment A B is 1 : 2 .
Explanation
Problem Setup We are given the coordinates of points A ( − 7 , − 14 ) , B ( 5 , 10 ) , and C ( − 3 , − 6 ) . We want to find the ratio in which C divides the line segment A B .
Calculate AC and CB We can use the distance formula to find the lengths of A C and CB . The distance formula between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
So, the length of A C is: A C = (( − 3 ) − ( − 7 ) ) 2 + (( − 6 ) − ( − 14 ) ) 2 = ( 4 ) 2 + ( 8 ) 2 = 16 + 64 = 80
And the length of CB is: CB = ( 5 − ( − 3 ) ) 2 + ( 10 − ( − 6 ) ) 2 = ( 8 ) 2 + ( 16 ) 2 = 64 + 256 = 320
Determine the Ratio Now we find the ratio A C : CB :
CB A C = 320 80 = 320 80 = 4 1 = 2 1
So the ratio is 1 : 2 .
Examples
Understanding ratios is crucial in various real-life scenarios. For instance, when mixing paint, a ratio helps determine the correct proportions of different colors to achieve the desired shade. Similarly, in baking, ratios ensure the right balance of ingredients for a perfect cake. In construction, ratios are used to calculate the correct mixture of cement, sand, and gravel for strong and durable concrete. These examples highlight how ratios are essential for achieving accurate and consistent results in everyday tasks.
Point C ( − 3 , − 6 ) divides segment ar{AB} , with endpoints A ( − 7 , − 14 ) and B ( 5 , 10 ) , in the ratio of 1 : 2 after calculating the distances of A C and CB . Therefore, the correct answer is 1 : 2 .
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