Given: KLMN is a trapezoid m∠N = m∠KML ME⊥KN, ME = 3√5, KE = 8, LM/KN = 3/5 Find: KM, LM, KN, Area of KLMN
To find the lengths of KM, LM, KN, and the area of trapezoid KLMN, we can use several properties and formulas of trapezoids.
1. Since we know that KLMN is a trapezoid, it means that the sides KL and MN are parallel.
2. Given that LM/KN = 3/5, we can determine the ratio of the lengths of LM and KN. Let's call the length of LM as x, and the length of KN as y. We can set up the equation:
x/y = 3/5
3. Since ME⊥KN, it means that the line segment ME is perpendicular to KN. This indicates that the triangles KMEN and KLEN are right triangles.
Now, let's find the values step by step:
Step 1: Finding KN
Since we know the length of KE (8) and ME (3√5), we can find KN using the Pythagorean theorem in triangle KMEN:
KE^2 + ME^2 = KN^2
8^2 + (3√5)^2 = KN^2
64 + 9*5 = KN^2
109 = KN^2
KN = √109
Step 2: Finding LM
Using the ratio obtained earlier: x/y = 3/5
Substituting the values we have, we get:
x/√109 = 3/5
Cross-multiplying: 5x = 3√109
Solving for x, we get:
x = (3√109)/5
Hence, LM = (3√109)/5
Step 3: Finding KM
Since LM/KN = 3/5, we can use this information to find KM:
KM = KN + LM
KM = √109 + (3√109)/5
KM = (√109 + 3√109)/5
KM = (4√109)/5
Step 4: Finding the Area of KLMN
The area of a trapezoid can be found using the formula:
Area = (base1 + base2) × height / 2
In our case, the bases are KL and MN, and the height is KE, which is the same as ME (since ME⊥KN).
Substituting the values, we get:
Area = (KL + MN) × KE / 2
Area = (KM + LM) × ME / 2
Area = ((4√109)/5 + (3√109)/5) × 3√5 / 2
Area = ((7√109)/5) × 3√5 / 2
Area = (21√109) / 10
Therefore, the area of trapezoid KLMN is (21√109) / 10.