Line segment AB with length a is divided by points P and Q into three line segments: AP , PQ, and QB , such that AP = 2PQ=2QB. Find:
A)the distance between point A and the midpoints of the segments the segment QB
B) the distance between the midpoints of segments AP and QB.
I don't get what this question wants from me can you please describe the proces and the steps for this question
Thanks
:)
Suppose QB has length x
Then PQ = x
AP = 2x
So, a = 4x
A) a/4 and 7a/8
B) 5a/8
Sure! Let's break down the steps to solve this question:
Step 1: Understand the given information
We are given that line segment AB is divided by points P and Q into three line segments: AP, PQ, and QB. We are also given that AP = 2PQ = 2QB.
Step 2: Draw the diagram
To better visualize the problem, draw a line segment AB and mark points P and Q on it. Also, label the distances AP, PQ, and QB based on the given information (AP = 2PQ and 2QB).
Step 3: Find the lengths of AP, PQ, and QB
To answer part A of the question (the distance between point A and the midpoints of the segment QB), we first need to find the length of QB. Since we are given that AP = 2PQ = 2QB, we can use this information to calculate the lengths.
Let's say the length of AP is x. Then, the length of PQ will be x/2, and the length of QB will be (x/2)/2 = x/4.
Step 4: Find the midpoint of the segment QB
To find the midpoint of the segment QB, we need to know the coordinates of points Q and B. If we assume that point Q is located at (0,0) on the number line, then point B would be located at (x/4,0).
To find the midpoint, we take the average of the x-coordinates and y-coordinates. The x-coordinate of the midpoint will be (0 + x/4)/2 = x/8.
Step 5: Calculate the distance between point A and the midpoint of QB
The distance between point A and the midpoint of QB can be found using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by the formula: sqrt((x2 - x1)^2 + (y2 - y1)^2).
In this case, x1 = 0, y1 = 0 (midpoint of QB), x2 = x (end point of AP), and y2 = 0.
So, the distance between point A and the midpoint of QB is sqrt((x - x/8)^2 + 0^2), which simplifies to sqrt(x^2/64).
Step 6: Calculate the distance between the midpoints of AP and QB
To answer part B of the question, we first need to find the midpoint of the segment AP. Following the same logic as in step 4, the x-coordinate of the midpoint of AP would be (x + x/2)/2 = 3x/8.
Now, we can calculate the distance between the midpoints of AP and QB. Using the distance formula, the distance is sqrt((3x/8 - x/8)^2 + 0^2), which simplifies to sqrt(x^2/64).
And there you have it! The distance between point A and the midpoints of the segment QB is sqrt(x^2/64), and the distance between the midpoints of segments AP and QB is also sqrt(x^2/64).