Question content area topPart 1Use the figure to the right to find the value of PT. T is the midpoint of PQ. PT 3x + 7 and TQ = 5x-9. PT = ?
Since T is the midpoint of PQ, this means that PT = TQ. Therefore, you can set the expressions for PT and TQ equal to each other and solve for x.
3x + 7 = 5x - 9
Rearrange to get 2x = 16
So x = 8
Substitute x = 8 into the equation for PT to get PT = 3(8) + 7 = 24 + 7 = 31
So PT = 31.
To find the value of PT, we need to use the information that T is the midpoint of PQ.
Let's use the formula for finding the midpoint of a line segment: M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 ).
In this case, we can find the midpoint T by using the coordinates of points P and Q.
Let's assume the coordinates of P are (x₁, y₁) and the coordinates of Q are (x₂, y₂).
Since T is the midpoint of PQ, we have the following equations:
x-coordinate of T = (x₁ + x₂) / 2
y-coordinate of T = (y₁ + y₂) / 2
Now, let's substitute the given information in the equations.
From the given information, PT = 3x + 7 and TQ = 5x - 9.
To find the value of PT, we can substitute the coordinates of points P and T into the distance formula:
PT = √[ (x₂ - x₁)² + (y₂ - y₁)² ]
Since T is the midpoint of PQ, the coordinates of T will be the average of the coordinates of P and Q:
x-coordinate of T = (x₁ + x₂) / 2
y-coordinate of T = (y₁ + y₂) / 2
Now, we can substitute the coordinates of T, P, and Q into the distance formula:
PT = √[ (x₂ - x₁)² + (y₂ - y₁)² ]
Since the y-coordinate of T is not given, we can only solve for the x-coordinate of T.
From the equation for the x-coordinate of T:
(x₁ + x₂) / 2 = x-coordinate of T
(x₁ + x₂) / 2 = 3x + 7
Multiply both sides by 2:
x₁ + x₂ = 6x + 14
From the equation for the y-coordinate of T:
(y₁ + y₂) / 2 = y-coordinate of T
(y₁ + y₂) / 2 = 5x - 9
We don't have enough information to solve for the value of PT. We need the y-coordinate of T or the equation of the line PQ to find the value of PT.
To find the value of PT, we can use the fact that T is the midpoint of PQ.
Given that PT = 3x + 7 and TQ = 5x - 9, we can set up an equation using the midpoint formula:
PT + TQ = PQ
Substituting the given values, we have:
3x + 7 + (5x - 9) = PQ
Combining like terms, we get:
8x - 2 = PQ
Since T is the midpoint of PQ, we know that PT = TQ. Therefore, PT = TQ = 5x - 9.
Setting PT equal to TQ, we have:
3x + 7 = 5x - 9
Next, we can solve the equation for x. Let's isolate the variable terms on one side:
3x - 5x = -9 - 7
Simplifying, we get:
-2x = -16
Now, divide both sides of the equation by -2 to solve for x:
x = (-16) / (-2)
x = 8
We have found the value of x to be 8. Now, we can substitute this value back into the expression for PT to find its value:
PT = 3(8) + 7
PT = 24 + 7
PT = 31
Therefore, the value of PT is 31.