What is the length of the the midsegment of a trapezoid with bases of length 19 and 29?
The length of the midsegment of a trapezoid is the average of the lengths of its bases.
Therefore, the length of the midsegment is: (19 + 29)/2 = 48/2 = 24.
Therefore, the length of the midsegment of the trapezoid is 24. Answer: \boxed{24}.
LMNP is a rectangle. Find the value of x and the length of each diagonal.
LN=2x+14 and MP=7x-6
Since LMNP is a rectangle, opposite sides are congruent. Therefore, LN = MP.
Given LN = 2x + 14 and MP = 7x - 6:
2x + 14 = 7x - 6
Adding 6 to both sides: 2x + 20 = 7x
Subtracting 2x from both sides: 20 = 5x
Dividing both sides by 5: x = 4
Therefore, x = 4.
To find the length of each diagonal, we can use the Pythagorean theorem.
Let's consider the diagonal LN. Its length would be the hypotenuse of a right triangle with LN, LM, and NM as its sides. Using the Pythagorean theorem:
LN^2 = LM^2 + NM^2
Substituting LN = 2x + 14 and NM = 7x - 6:
(2x + 14)^2 = LM^2 + (7x - 6)^2
Expanding both sides:
4x^2 + 56x + 196 = LM^2 + 49x^2 - 84x + 36
Combining like terms:
4x^2 + 56x + 196 = 49x^2 - 84x + 36
Subtracting 4x^2 and 49x^2 from both sides:
56x + 196 = -35x^2 - 84x + 36
Combining like terms:
35x^2 + 140x + 160 = 0
Dividing every term by 5:
7x^2 + 28x + 32 = 0
Now, we can factor the quadratic equation:
(7x + 4)(x + 8) = 0
Setting each factor equal to zero:
7x + 4 = 0 or x + 8 = 0
If 7x + 4 = 0, then x = -4/7.
If x + 8 = 0, then x = -8.
However, in this case, x cannot be negative since it represents a length. Therefore, x = -4/7 is extraneous.
Therefore, x = 8.
Now, substituting x = 8 back into LN = 2x + 14:
LN = 2(8) + 14 = 16 + 14 = 30.
Therefore, the length of each diagonal is LN = MP = 30. Answer: \boxed{30}.