Determine whether QS || PT . Justify your answer.
PQ = 18, QR = 6, RS = 12, ST = 36
To determine whether QS is parallel to PT, we need to compare the slopes of both line segments.
First, let's find the slopes of QS and PT. The slope of a line can be found using the formula:
slope = (change in y) / (change in x)
For QS:
Q(x₁, y₁) = (PQ, 0) = (18, 0)
S(x₂, y₂) = (36, 12)
slopeQS = (y₂ - y₁) / (x₂ - x₁)
= (12 - 0) / (36 - 18)
= 12 / 18
= 2 / 3
For PT:
P(x₁, y₁) = (0, 0)
T(x₂, y₂) = (18 + 6 + 12, 0) = (36, 0)
slopePT = (y₂ - y₁) / (x₂ - x₁)
= (0 - 0) / (36 - 0)
= 0 / 36
= 0
Now, we compare the slopes:
slopeQS = 2 / 3
slopePT = 0
Since the slopes are different (2/3 ≠ 0), QS and PT are not parallel.
To determine whether QS is parallel to PT, we can use the property that if the ratios of corresponding side lengths in two triangles are equal, then the two triangles are similar. Therefore, we need to compare the ratios of corresponding side lengths in triangle PQS and triangle PRT.
Let's start by calculating the ratios of corresponding side lengths in triangle PQS and triangle PRT.
In triangle PQS:
PQ/QS = 18/QS
QR/RS = 6/12
PS/SQ = PQ/RS = 18/12
In triangle PRT:
PT/TR = ?
QR/RS = 6/12
PR/RT = PQ/RS = 18/12
Now, let's simplify the ratios:
PQ/QS = 18/QS
QR/RS = 6/12 = 1/2
PS/SQ = 18/12 = 3/2
PT/TR = ?
QR/RS = 6/12 = 1/2
PR/RT = 18/12 = 3/2
By comparing the ratios of corresponding side lengths in both triangles, we can see that PQ/QS = PT/TR, QR/RS is equal to QR/RS, and PS/SQ = PR/RT.
Since all three ratios are equal, we can conclude that QS is parallel to PT. The justification is that the corresponding side length ratios in both triangles are equal, which satisfies the property of similarity.