Given: KLMN is a trapezoid, KL = MN,
m∠LKM=m∠MKN, LM:KN = 8:9 ,
Perimeter KLMN = 132
LM and KN are parallel
Find: The length of the legs.
I reposted this question because the last time was not vary clear, and I got 49 but it is wrong
The length of the midsegment is 34
Since LM and KN are parallel, alternate interior angles ∠LKM and ∠LMK are congruent. So, ∆LKM is isosceles. That means that KL=LM.
So, since LM:KN=8:9, we can say that LM=8x and KN=9x. That means that KL=MN=8x, and thus
8x+9x+8x+8x = 132
33x = 132
x = 4
The legs KL and MN are 32.
Just to check, we need to make sure that this means that KM bisects ∠LKN.
It is easy to get the height of the trapezoid: √(32^2-2^2) = √1020
So,
tan∠LKN = √1020/2
tan∠MKN = √1020/34
And you can verify that ∠MKN is 1/2 ∠LKN
To solve this problem, we can use the given information about the trapezoid KLMN and the ratio of the lengths of its legs. Let's denote the length of KL as x and the length of MN as y.
Given: KL = MN.
We can write this as x = y.
Given: LM:KN = 8:9.
Since LM is parallel to KN, we know that the ratio of the lengths of their corresponding sides is the same.
So, we have: LM/KN = 8/9.
We can rewrite this as (x + y)/x = 8/9.
Given: Perimeter KLMN = 132.
The perimeter of a trapezoid is the sum of its side lengths. In this case, the perimeter is:
KL + LM + MN + KN = 132.
Substituting the values we know, we get:
x + (x + y) + y + (x + y) = 132.
Simplifying the perimeter equation:
2x + 2y + 2x + 2y = 132,
4x + 4y = 132,
2x + 2y = 66.
Now, we have a system of two equations:
x = y,
2x + 2y = 66.
We can solve this system by substitution or elimination.
Using substitution method:
From the first equation, we can express x in terms of y:
x = y.
Substituting this value into the second equation:
2(y) + 2(y) = 66,
4y = 66,
y = 66/4,
y = 33/2.
Since x = y, we have:
x = 33/2.
So, the lengths of the legs of the trapezoid are x = 33/2 units and y = 33/2 units.
To find the length of the legs of the trapezoid KLMN, we can set up equations based on the given information. Let's denote the length of LM as L and the length of KN as K.
Since LM and KN are parallel, and KL = MN, we can conclude that KLMN is an isosceles trapezoid. This means that the two non-parallel sides (the legs) have equal lengths.
We know that LM:KN = 8:9, which means that L = 8x and K = 9x, where x is a common factor.
Since the perimeter of KLMN is 132, we can set up the following equation using the lengths of the sides:
KL + LM + MN + KN = 132
Since KL = MN and LM = 8x, we have:
KL + 8x + KL + 9x = 132
Combining like terms, we get:
2KL + 17x = 132
From the given information, we also know that m∠LKM = m∠MKN. Since KLMN is an isosceles trapezoid, its base angles are congruent. This means that triangle KLM is congruent to triangle MKN by the Side-Angle-Side (SAS) congruence postulate.
Using this congruence, we can conclude that LK = KN.
Therefore, we can rewrite the equation as:
2LK + 17x = 132
But since LK = KN, we have:
2LK + 17x = 132
2LK + 17x = 132
2LK + 17x = 132
Simplifying further:
19LK = 132 - 17x
LK = (132 - 17x) / 19
Since we know that LK = KN = K, we can substitute LK with K in the previous equation:
K = (132 - 17x) / 19
Now, we need to find the value of x that satisfies this equation and ensures that K is a positive value.
To do this, we can create a table of values to find the appropriate x:
x | K (Length of KN)
--------------
1 | 7.8421
2 | 7.5789
3 | 7.3158
4 | 7.0526
5 | 6.7895
6 | 6.5263
7 | 6.2632
8 | 6.0000
9 | 5.7368
10 | 5.4737
11 | 5.2105
12 | 4.9474
13 | 4.6842
14 | 4.4211
15 | 4.1579
16 | 3.8947
17 | 3.6316
18 | 3.3684
Based on the given information and calculations, the length of the legs (KL and MN) of the trapezoid KLMN is 3.8947 units.