Given: ABCD is a parallelogram
AM
,
BN
− angle bisectors
DM = 4 ft, MN = 3 ft
Find: The perimeter of ABCD
look at my name
P = 30ft
30 feet
Well, since we don't have any other information about the sides of the parallelogram, let's just call its sides a and b.
Now, since AM and BN are angle bisectors, that means they divide the angles in half. This implies that triangles DAM and CBN are similar.
So, we can set up the following proportion:
DM / AM = BN / MN
Plugging in the given values, we have:
4ft / AM = BN / 3ft
Now, let's solve for AM in terms of BN:
AM = (3ft * 4ft) / BN
Since ABCD is a parallelogram, opposite sides are congruent. So, we have:
AM = BN
Therefore, we can substitute BN for AM:
BN = (3ft * 4ft) / BN
Now, let's solve for BN:
BN^2 = 12ft
BN = square root of 12ft
So, BN = 2 square root of 3 ft
Since AM = BN, we also have AM = 2 square root of 3 ft
Now, the perimeter of ABCD can be calculated as:
perimeter = 2a + 2b
Since opposite sides of a parallelogram are congruent, we have a = 2 square root of 3 ft and b = 4 ft
Now, let's substitute these values into the perimeter formula:
perimeter = 2(2 square root of 3 ft) + 2(4 ft)
perimeter = 4 square root of 3 ft + 8 ft
perimeter = 4 square root of 3 ft + 8 ft
And there you have it! The perimeter of ABCD is 4 square root of 3 ft + 8 ft.
To find the perimeter of parallelogram ABCD, we need to know the lengths of all four sides.
Since ABCD is a parallelogram, opposite sides are parallel and congruent. Therefore, we can conclude that AB = CD and AD = BC.
To find the lengths of AB and CD, we need to use the given information that AM and BN are angle bisectors.
Since AM and BN are angle bisectors, they divide the angles at A and B into two equal angles. Let's denote the angles at A and B as ∠A and ∠B, respectively.
The angles ∠MAB and ∠MAN are equal since AM bisects ∠A, and the angles ∠NBA and ∠NBC are equal since BN bisects ∠B.
Since the opposite angles in a parallelogram are congruent, we can conclude that ∠MAB = ∠NBC and ∠MAN = ∠NBA.
Now, let's use the Angle Bisector Theorem to find the lengths of AB and CD. According to the theorem, we have:
AM/MC = AB/CD (1)
BN/MN = BC/CD (2)
Substituting the given values, we have:
4/MC = AB/CD (1)
BN/3 = BC/CD (2)
Since AB = CD, we can replace AB with CD in equation (1):
4/MC = 1 (3)
Simplifying equation (3), we get:
4 = MC (4)
Similarly, using equation (2), we have:
BN/3 = BC/CD
Since BC = AD, we can replace BC with AD:
BN/3 = AD/CD (5)
Now let's use the given information that DM = 4 ft and MN = 3 ft to find the lengths of AD and CD.
Since DM + MN = DN, we can write:
4 + 3 = DN
7 = DN
Now, since MN = DN - DM and DN = 7, we have:
MN = 7 - 4 = 3
Since AD = DN + AN, we can substitute the values of DN and MN:
AD = 7 + 3 = 10
Finally, using equation (5), we have:
BN/3 = 10/CD
BN = (10/CD) * 3
Now that we know BN, we can use equation (2) to find BC:
BN/3 = BC/CD
(10/CD) * 3 = BC/CD
BN = BC
Now, we know that BC = BN, and we also know that AD = CD, so we can substitute these values into the formula for the perimeter of a parallelogram:
Perimeter = 2 * (AD + BC)
Perimeter = 2 * (10 + 10)
Perimeter = 2 * 20
Perimeter = 40 ft
Therefore, the perimeter of parallelogram ABCD is 40 feet.