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In the figure above, the length of AD is 15. Which of the following st
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21 Jun 2023, 05:28
21 Jun 2023, 05:28
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In the figure above, the length of AD is 15. Which of the following statements individually provide(s) sufficient additional information to determine the area of parallelogram above?
Indicate all that apply
A. Distance between AB and DC
B. Shortest distance between BC and AD is 4
C. AC = 10
D. Area of triangle AOD is 30
E. Triangle ABD is equilateral triangle
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Re: In the figure above, the length of AD is 15. Which of the following st
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27 Jun 2023, 06:51
27 Jun 2023, 06:51
1
How does "B. Shortest distance between BC and AD is 4" provide enough info to determine the area of the parallelogram?
If the shortest distance between BC and AD IS a line perpendicular to them, then yes, it is enough info because the shortest distance is the height of the parallelogram.
However, if the shortest distance between BC and AD is NOT a line perpendicular to them, then how is it enough info? For example:
- Point A has coordinates (0, 0)
- Point D has coordinates (15, 0)
- Point B has coordinates (15 + sqrt(15), 1)
- Point C has coordinates (30 + sqrt(15), 1)
In this case, the shortest distance between BC and AD is the line between point B and D, which is 4.
In the example above, we wouldn't know that the height of the parallelogram is 1.
If the shortest distance between BC and AD IS a line perpendicular to them, then yes, it is enough info because the shortest distance is the height of the parallelogram.
However, if the shortest distance between BC and AD is NOT a line perpendicular to them, then how is it enough info? For example:
- Point A has coordinates (0, 0)
- Point D has coordinates (15, 0)
- Point B has coordinates (15 + sqrt(15), 1)
- Point C has coordinates (30 + sqrt(15), 1)
In this case, the shortest distance between BC and AD is the line between point B and D, which is 4.
In the example above, we wouldn't know that the height of the parallelogram is 1.
Re: In the figure above, the length of AD is 15. Which of the following st
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27 Jun 2023, 08:20
27 Jun 2023, 08:20
Expert Reply
OE
Area of parallelogram = base × perpendicular height We know base here, as AD = 15 To calculate the area, we need to know its perpendicular height.
Let us find out, which of the answer choices would be sufficient to calculate its perpendicular height.
A) It says, if we know the distance between AB and DC, which will represent the perpendicular distance, in that case, the base needs to be the length CD, and as we do not know the length CD, so we cannot find area. Hence this statement is not sufficient to determine area of parallelogram.
B) Shortest distance between BC and AD is 4. Shortest distance is nothing but perpendicular distance between BC and AD.
So, here we know base as well as perpendicular height. So, we can calculate the area of parallelogram.
C) By knowing the distance of AC, we cannot find the perpendicular distance between AD and BC. So, it is not sufficient to determine the area.
D) Area of triangle AOD = 30,In a parallelogram, Diagonals do bisect each other and opposite sides are equal. So, we can say, Area of triangle AOD is equal to the area of triangle BOC.
Height of parallelogram ABCD, is twice the height of triangle AOD.
It is given that, Area of triangle AOD = 30
1/2× 12 × h = 30
h = 5 units. So, height of parallelogram is 10units.
We know height and base of parallelogram ABCD, hence we can calculate its area.
E) If triangle ABD is an equilateral triangle. Height of triangle ABD would be the height of parallelogram.
We know the height of equilateral triangle \(\frac{\sqrt{3}}{2} \times side = \frac{\sqrt{3}}{2} \times 15= \frac{15 \sqrt{3}}{2}\)
Through this data, we know base of parallelogram (15 units) and height of parallelogram \( \frac{15 \sqrt{3}}{2}\)
Hence, we can calculate the area of parallelogram.
Ans. (B, D, E)
Area of parallelogram = base × perpendicular height We know base here, as AD = 15 To calculate the area, we need to know its perpendicular height.
Let us find out, which of the answer choices would be sufficient to calculate its perpendicular height.
A) It says, if we know the distance between AB and DC, which will represent the perpendicular distance, in that case, the base needs to be the length CD, and as we do not know the length CD, so we cannot find area. Hence this statement is not sufficient to determine area of parallelogram.
B) Shortest distance between BC and AD is 4. Shortest distance is nothing but perpendicular distance between BC and AD.
So, here we know base as well as perpendicular height. So, we can calculate the area of parallelogram.
C) By knowing the distance of AC, we cannot find the perpendicular distance between AD and BC. So, it is not sufficient to determine the area.
D) Area of triangle AOD = 30,In a parallelogram, Diagonals do bisect each other and opposite sides are equal. So, we can say, Area of triangle AOD is equal to the area of triangle BOC.
Height of parallelogram ABCD, is twice the height of triangle AOD.
It is given that, Area of triangle AOD = 30
1/2× 12 × h = 30
h = 5 units. So, height of parallelogram is 10units.
We know height and base of parallelogram ABCD, hence we can calculate its area.
E) If triangle ABD is an equilateral triangle. Height of triangle ABD would be the height of parallelogram.
We know the height of equilateral triangle \(\frac{\sqrt{3}}{2} \times side = \frac{\sqrt{3}}{2} \times 15= \frac{15 \sqrt{3}}{2}\)
Through this data, we know base of parallelogram (15 units) and height of parallelogram \( \frac{15 \sqrt{3}}{2}\)
Hence, we can calculate the area of parallelogram.
Ans. (B, D, E)
