EF is a median of isosceles trapezoid ABCD. Find EF if AB is equal to 3 inches, and CD is equal to 11 inches.
Wouldn't it be (11+3)/2 ?
EF is a median of isosceles trapezoid ABCD. Find EF if AB is equal to 3 inches, and CD is equal to 11 inches.
To find EF, we need to determine the length of the median EF in the isosceles trapezoid ABCD.
In an isosceles trapezoid, the diagonals (AC and BD) are congruent and they intersect at point E. So, EF is congruent to BD.
Since AB and CD are parallel sides of the trapezoid, BD can be determined by subtracting the lengths of AB and CD from the sum of all four sides:
BD = (AB + BC + CD) - (AB + CD)
BD = BC
Therefore, we need to find the length of BC.
In an isosceles trapezoid, the bases (AB and CD) are parallel and congruent. So, BC is equal to the average of AB and CD:
BC = (AB + CD) / 2
Substituting the given values:
BC = (3 + 11) / 2
BC = 14 / 2
BC = 7 inches
Now that we know BC, we can determine the length of EF:
EF = BD = BC = 7 inches
Therefore, EF is equal to 7 inches.
To find the length of EF, we can use the properties of isosceles trapezoid ABCD. Let's start by drawing the trapezoid and labeling the given information:
A _________ B
| |
| |
|_________|
D C
Given: AB = 3 inches and CD = 11 inches.
Since EF is a median of the trapezoid, it will divide the trapezoid into two congruent triangles. Let's label the point where EF intersects AD as M, and the point where EF intersects BC as N.
A _________ B
| \ | / |
| | |
| | |
D ____|____ C
M N
We know that a median divides a triangle into two congruent triangles. So, triangle AME is congruent to triangle BMF, and triangle CNE is congruent to triangle DNF.
Now, let's consider triangle AME. Since EF is a median, AM is congruent to BM. Since ABCD is an isosceles trapezoid, base AD is parallel to base BC, resulting in AM being congruent to BN as well.
Therefore, we can conclude that AM = BM = BN.
Now, let's consider triangle AME. We have AM = BN and AE (which is half of AB) = 3/2 inches.
Since AE = BN, we can set up the following equation:
3/2 inches + BN + NE = CD = 11 inches
Simplifying the equation, we get:
3/2 inches + 2BN = 11 inches
Subtracting 3/2 inches from both sides, we get:
2BN = 11 inches - 3/2 inches
2BN = 22/2 inches - 3/2 inches
2BN = 19/2 inches
Dividing both sides by 2, we get:
BN = (19/2) / 2
BN = 19/4 inches
Since BN is half of EF, we can calculate EF by multiplying BN by 2:
EF = BN * 2
EF = (19/4) * 2
EF = 19/2 inches
Therefore, the length of EF is 19/2 inches.