Answer:
a) [tex]x=4[/tex] b) [tex]PT = 13, TQ = 13, PQ=26[/tex]
Step-by-step explanation:
Hi! I hope you had a good New Year! Ok, so we need to find the value of X and the measures of PT, TQ, and PQ.
On the top right, we are given a line with the points P, T, and Q on it.
But, the most important information that we are given is that T is the midpoint PQ.
A midpoint is, well a point, that splits a line segment into two equal parts. It splits a line segment completely in half.
In this case, the midpoint T splits the line segment PQ into two congruent line segments.
Those two congruent line segments are TQ and PQ.
We don't have a numerical value for the measurements TQ and PQ yet, but we are given the measurements of them based on algebraic expressions, or in terms of x.
Looking at the digaram to the right, we see that:
[tex]TQ = 5x-7[/tex]
[tex]PT = 3x+1[/tex]
We said that TQ and PT are congruent segments because of the definition of a midpoint. If they are congruent to each other, then they must have an equal measurement. This means that we can set the measurements of TQ and PT equal to each other.
So instead of TQ = PT, we can say this now:
[tex]5x-7 = 3x+1[/tex]
See how this equation helps us to solve for x?
So, this is an equation with x's on both sides, but we want x is to be just on one side. There are a few ways to solve for x, but I'm just going to have x on the left side.
In order to have x on the left side, I need to subtract both sides by 3x.
The x's will cancel out on the right side.
On the left side, I'll have 5x - 3x, which simplifies to 2x.
So, we have this now:
[tex]2x-7=1[/tex]
Ok, I want 2x to be completely alone on the left side. To be able to do that, I am going to add both sides by 7.
On the left side, the 7's will cancel out.
On the right side, we have 1 + 7, which is 8.
After doing that, we have:
[tex]2x=8[/tex]
The final step is to divide both sides by 2.
On the left side, the 2's will cancel out.
On the right side, we have 8 divided by 2, which equals 4.
Therefore, x = 4.
Great! Now that we know what the value of x is, we can use it to find the measures of PT, TQ, and PQ.
Remember that:
TQ = [tex]5x-7[/tex]
PT = [tex]3x+1[/tex]
Initially, we didn't know what the numerical measures of TQ and PT were because they had x in them and we did not know what the value of x was at the time.
However, since we know that x = 4 now, we can just plug in those values for x into the equations. Let's do TQ
So, TQ = [tex]5(4) - 7[/tex]. Note: 5(4) means 5 times 4.
5 times 4 equals 20. 20 -7 = 13
Great! So, TQ = 13
Let's find the measurement for PT now.
PT = [tex]3(4)+1[/tex] = [tex]12+1[/tex] = [tex]13[/tex]
So, PT = 13.
This makes sense because we said that PT = TQ have the same measurement because of the definition of the midpoint, so they must both equal the same number, which is 13 in this case.
To find PQ, we just need to add the two parts, PT and PQ. Thankfully, we know what the values of PT and PQ are.
So,
[tex]PQ = PT + TQ[/tex]
[tex]PQ = 13+13 = 26[/tex]
So, PQ = 26.
I hope this helps! If there is any part of my explanation that does not make sense, please let me know! I can always clarify if needed. Have a great rest of your day!
