Given △QRS ≅△TUV , QS = 4w+16 and TV = 6w - 8, find the length of TV and QS. Show all the steps that you used to solve this problem in order to earn full credit. (2 points)
If △QRS ≅△TUV, then we know that the corresponding sides are proportional. Therefore, we can set up the following equation:
QS/TV = (4w+16)/(6w-8)
To solve for QS and TV, we can cross multiply:
QS * (6w-8) = TV * (4w+16)
Expanding this equation, we get:
6w * QS - 8 * QS = 4w * TV + 16 * TV
Simplifying further, we have:
6wQS - 8QS = 4wTV + 16TV
Now, let's solve for QS. We can factor out QS on the left-hand side:
QS(6w - 8) = 4wTV + 16TV
Dividing both sides by (6w - 8), we get:
QS = (4wTV + 16TV)/(6w - 8)
To find the length of QS, we need to know the length of TV. However, we are not given the value of w, so we cannot solve for TV and QS at this time.
To find the lengths of QS and TV, we need to solve for the values of w first.
Since △QRS is congruent to △TUV, we can use congruence properties to find a relationship between the corresponding sides.
Given: QS = 4w + 16 and TV = 6w - 8.
Using the congruent triangles, we know that the corresponding sides are equal. Therefore, we have:
QS = TV
Substituting the values we know:
4w + 16 = 6w - 8
Rearranging the equation to isolate the variable:
4w - 6w = -8 - 16
-2w = -24
Dividing both sides by -2:
w = -24 / -2
w = 12
Now that we have found the value of w, we can substitute it back into the given expressions to find the lengths of QS and TV.
QS = 4w + 16
QS = 4(12) + 16
QS = 48 + 16
QS = 64
Therefore, the length of QS is 64.
TV = 6w - 8
TV = 6(12) - 8
TV = 72 - 8
TV = 64
Therefore, the length of TV is 64.
So, the length of TV is 64 and the length of QS is also 64.