Question

У паралелограмі АВСD висота ВЕ ділить сторону AD навпіл. Обчислити сторону АВ, якщо периметр паралелограма дорівнює 7, а периметр трикутника АВD дорівнює 5.

Answer 1

2(AB+AD)=7⇒AB+AD=3,5
BH_|_AD U AH=DH⇒BH-медиана и высота⇒ΔABD-равнобедренный⇒AB=BD
AB+BD+AD=2AB+AD=5
3,5+AB=5
AB=5-3,5
AB=1,5

Answer 2

Given Information

We are given a parallelogram ABCD with height BE dividing side AD in half. The perimeter of the parallelogram is 7, and the perimeter of triangle ABD is 5.

Solution

To find the length of side AB, we can use the given information about the perimeters of the parallelogram and the triangle.

Let's denote the length of side AB as x.

The perimeter of the parallelogram is the sum of the lengths of all four sides, which is 7. Since opposite sides of a parallelogram are equal in length, we can write:

2(AB + BC) = 7

Similarly, the perimeter of the triangle ABD is the sum of the lengths of its three sides, which is 5. Since side AB is shared by both the parallelogram and the triangle, we can write:

AB + BD + AD = 5

Since the height BE divides side AD in half, we can write:

AD = 2BE

Now, let's substitute the value of AD in terms of BE into the equation for the perimeter of the triangle:

AB + BD + 2BE = 5

Since BD is equal to BC (opposite sides of a parallelogram are equal), we can simplify the equation:

AB + 2BC + 2BE = 5

Now, let's substitute the value of BC in terms of AB into the equation for the perimeter of the parallelogram:

2(AB + AB/2) + 2BE = 7

Simplifying this equation, we get:

3AB + 2BE = 7

Now, we have a system of two equations:

3AB + 2BE = 7 (Equation 1) AB + 2BC + 2BE = 5 (Equation 2)

We can solve this system of equations to find the values of AB and BE.

Let's solve Equation 2 for BC:

BC = (5 - AB - 2BE)/2

Now, substitute this value of BC into Equation 1:

3AB + 2BE = 7

3AB + 2(5 - AB - 2BE)/2 = 7

Simplifying this equation, we get:

3AB + 5 - AB - 2BE = 7

2AB - 2BE = 2

AB - BE = 1 (Equation 3)

Now, let's solve Equation 3 for AB:

AB = BE + 1

Since we know that the height BE divides side AD in half, we can substitute AD = 2BE into the equation:

AB = 2BE + 1

Now, let's substitute the value of AB in terms of BE into Equation 1:

3(2BE + 1) + 2BE = 7

Simplifying this equation, we get:

6BE + 3 + 2BE = 7

8BE = 4

BE = 0.5

Now, substitute the value of BE into the equation for AB:

AB = 2(0.5) + 1

AB = 2 + 1

AB = 3

Therefore, the length of side AB is 3.

Answer

The length of side AB in the parallelogram ABCD is 3.