in the figure ABCD is a parallelogram. AL is perpendicular to BC, AM is perpendicular to CD , AL = 4 cm and AM = 5 cm if BC =6.5 cm then find the value of CD
To find the value of CD, we will use the properties of a parallelogram and the given information.
Let's start by considering the given information:
1. AB || CD (given that ABCD is a parallelogram)
2. AL ⊥ BC (given that AL is perpendicular to BC)
3. AM ⊥ CD (given that AM is perpendicular to CD)
4. AL = 4 cm
5. AM = 5 cm
6. BC = 6.5 cm
From property 2, we know that AL and BC are perpendicular, which means AL is the height of the parallelogram.
Similarly, we can conclude that AM is the height of the parallelogram, since it is perpendicular to CD (property 3).
Since AL is the height of the parallelogram and AL = 4 cm, we can conclude that the area of the parallelogram is equal to the length of BC multiplied by the height, which is 6.5 cm (BC) multiplied by 4 cm (AL).
Area of parallelogram = BC × AL
On substituting the values, we have:
Area of parallelogram = 6.5 cm × 4 cm = 26 cm²
Now, to find CD, we can use the fact that the area of a parallelogram is equal to the length of the base multiplied by the height. Since AM is the height of the parallelogram and AM = 5 cm, we can write:
Area of parallelogram = CD × AM
On substituting the values, we have:
26 cm² = CD × 5 cm
Now, to find the value of CD, we can solve the equation:
26 cm² = CD × 5 cm
To isolate CD, divide both sides of the equation by 5 cm:
CD = 26 cm² / 5 cm
Calculating this value:
CD = 5.2 cm
Therefore, the value of CD is 5.2 cm.
Had to experiment until I got your figure correct, I think I got it
given: BC = AD = 6.5
angle B = angle D = Ø
in triangle AMD,
sinD = sinØ
= 5/6.5
D = appr 50.28° (I stored it in my calculator)
In triangle ABL
sinB = sinØ
= 4/AB
5/6.5 = 4/AB
5AB = 26
AB = 26/5 = 5.2
since CD = AB
CD = 5.2
At this point I realize I could have skipped all the above and just used the ratio of our two similar triangles and started with
5/6.5 = 4/AB