Final answer:
To find the coordinates of point C, we can use the given ratio AC:AB. The x-coordinate of C is 17 and the y-coordinate is 26. The statement that is not true is d) The ratio of the x-coordinates of C to A is 3:8.
Step-by-step explanation:
To find the coordinates of point C, we can use the ratio AC:AB, which is given as 3:8. This means that if we divide the distance between A and B into 8 equal parts, point C would be located at 3 of those parts.
To calculate the x-coordinate of point C, we find the difference between the x-coordinates of A and B (31-15 = 16) and divide it by 8. Multiplying this result by 3 and adding it to the x-coordinate of A (15), we get the x-coordinate of C as 17.
To calculate the y-coordinate of point C, we do the same process using the y-coordinates of A and B. The difference between the y-coordinates of A and B (52-20 = 32) divided by 8 gives us the length of one part. Multiplying this length by 3 and adding it to the y-coordinate of A (20), we get the y-coordinate of C as 26.
From the coordinates of point C (17, 26), we can evaluate each of the given statements.
(a) The x-coordinate of point C is greater than 15. This statement is true since 17 is greater than 15.
(b) The y-coordinate of point C is less than 52. This statement is true since 26 is less than 52.
(c) The distance from A to C is three times the distance from C to B. This statement is true since we can use the distance formula to calculate the distances and verify this relationship.
(d) The ratio of the x-coordinates of C to A is 3:8. This statement is false because the ratio should be the other way around, 8:3.
(e) The ratio of the y-coordinates of C to B is 3:8. This statement is true since the y-coordinate of C is 26 and the y-coordinate of B is 52, which gives us the ratio of 26:52, which simplifies to 1:2, and when multiplied by 3, we get 3:6, which is equivalent to 3:8.