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Given ΔQRS≅ΔTUV
, QS=3v+2
and TV=7v−6
, find the length of QS and TV.
(2 points)
Since ΔQRS ≅ ΔTUV, we know that their corresponding sides are proportional.
Using this information, we can set up the equation:
QS/TV = (3v+2)/(7v-6)
To find the length of QS and TV, we need to solve for v.
Cross-multiplying, we have:
QS * (7v-6) = TV * (3v+2)
Expanding both sides of the equation:
7vQS - 6QS = 3vTV + 2TV
Rearranging the equation:
7vQS - 3vTV = 6QS + 2TV
Factoring out the v on the left side:
v(7QS - 3TV) = 6QS + 2TV
Dividing both sides by (7QS - 3TV):
v = (6QS + 2TV) / (7QS - 3TV)
To find the length of QS and TV, we substitute v back into QS=3v+2 and TV=7v-6.
For QS:
QS = 3v + 2
QS = 3[(6QS + 2TV) / (7QS - 3TV)] + 2
Multiplying both sides of the equation by (7QS - 3TV):
QS(7QS - 3TV) = 3(6QS + 2TV) + 2(7QS - 3TV)
Expanding both sides:
7QS^2 - 3QSTV = 18QS + 6TV + 14QS - 6TV
Simplifying:
7QS^2 - 3QSTV = 32QS
Rearranging the equation:
7QS^2 - 32QS - 3QSTV = 0
For TV:
TV = 7v - 6
TV = 7[(6QS + 2TV) / (7QS - 3TV)] - 6
Multiplying both sides of the equation by (7QS - 3TV):
TV(7QS - 3TV) = 7(6QS + 2TV) - 6(7QS - 3TV)
Expanding both sides:
7QSTV - 3TV^2 = 42QS + 14TV - 42QS + 18TV
Simplifying:
7QSTV - 3TV^2 = 32TV
Rearranging the equation:
7QSTV - 32TV - 3TV^2 = 0
Now, we have two quadratic equations. We can solve these equations to find the lengths of QS and TV.
To find the length of QS and TV, we need to set up an equation using the given information that ΔQRS≅ΔTUV.
Since the two triangles are congruent, the corresponding sides are equal in length. Therefore, we have:
QS = TV
Now, using the given lengths of QS and TV, we can substitute them into the equation:
3v + 2 = 7v - 6
To solve this equation, we can start by isolating the variable v on one side:
3v - 7v = -6 - 2
-4v = -8
Next, we can divide both sides of the equation by -4 to solve for v:
v = (-8) / (-4)
v = 2
Now that we have found the value of v, we can substitute it back into the equation QS = TV to find the lengths of QS and TV:
QS = 3v + 2
QS = 3(2) + 2
QS = 6 + 2
QS = 8
TV = 7v - 6
TV = 7(2) - 6
TV = 14 - 6
TV = 8
Therefore, the length of QS is 8 and the length of TV is also 8.
To find the lengths of QS and TV, we need to equate the corresponding sides of the congruent triangles ΔQRS and ΔTUV.
Given:
ΔQRS is congruent to ΔTUV.
QS = 3v + 2
TV = 7v - 6
We know that corresponding sides of congruent triangles are equal in length. Therefore, we can equate the corresponding sides:
QS = TV
Substituting the given values for QS and TV, we have:
3v + 2 = 7v - 6
Now, we can solve this equation for the value of v.
Let's isolate the variable on one side of the equation:
3v - 7v = -6 - 2
-4v = -8
Dividing both sides by -4 to solve for v:
v = (-8) / (-4)
v = 2
Now that we have the value of v, we can substitute it back into the original equations to find the lengths of QS and TV:
QS = 3v + 2
QS = 3(2) + 2
QS = 6 + 2
QS = 8
TV = 7v - 6
TV = 7(2) - 6
TV = 14 - 6
TV = 8
Therefore, the length of QS is 8 units and the length of TV is also 8 units.