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In a quadrilateral ABCD, the length of diagonal AC is 28 cm. The length of perpendicular from B and D on AC are 7 cm and 9 cm respectively. Then, the area of ABCD is:
- a)264
- b)224
- c)324
- d)214
- e)None of the above
Correct answer is option 'B'. Can you explain this answer?
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In a quadrilateral ABCD, the length of diagonal AC is 28 cm. The leng...
Given:
- Quadrilateral ABCD
- Length of diagonal AC is 28 cm
- Length of perpendicular from B on AC is 7 cm
- Length of perpendicular from D on AC is 9 cm
To find:
- Area of quadrilateral ABCD
Solution:
Let's consider the perpendicular from B on AC as BE and the perpendicular from D on AC as DF.
Step 1: Find the length of BD
- In triangle ABC, using Pythagoras theorem, we can find the length of AB.
- AB^2 = BC^2 + AC^2
- AB^2 = 7^2 + 28^2
- AB^2 = 49 + 784
- AB^2 = 833
- AB ≈ 28.87 cm
- In triangle ACD, using Pythagoras theorem, we can find the length of AD.
- AD^2 = CD^2 + AC^2
- AD^2 = 9^2 + 28^2
- AD^2 = 81 + 784
- AD^2 = 865
- AD ≈ 29.41 cm
- Now, in triangle ABD, using Pythagoras theorem, we can find the length of BD.
- BD^2 = AB^2 + AD^2
- BD^2 = 28.87^2 + 29.41^2
- BD^2 = 834.3369 + 866.3281
- BD^2 ≈ 1700.665
- BD ≈ 41.23 cm
Step 2: Find the area of quadrilateral ABCD
- Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ACD
- Area of triangle ABC = 1/2 * AB * BE
- Area of triangle ABC = 1/2 * 28.87 * 7
- Area of triangle ABC ≈ 101.045 cm^2
- Area of triangle ACD = 1/2 * AD * DF
- Area of triangle ACD = 1/2 * 29.41 * 9
- Area of triangle ACD ≈ 132.345 cm^2
- Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ACD
- Area of quadrilateral ABCD ≈ 101.045 + 132.345
- Area of quadrilateral ABCD ≈ 233.39 cm^2
Therefore, the area of quadrilateral ABCD is approximately 233.39 cm^2, which is closest to option B (224 cm^2).
- Quadrilateral ABCD
- Length of diagonal AC is 28 cm
- Length of perpendicular from B on AC is 7 cm
- Length of perpendicular from D on AC is 9 cm
To find:
- Area of quadrilateral ABCD
Solution:
Let's consider the perpendicular from B on AC as BE and the perpendicular from D on AC as DF.
Step 1: Find the length of BD
- In triangle ABC, using Pythagoras theorem, we can find the length of AB.
- AB^2 = BC^2 + AC^2
- AB^2 = 7^2 + 28^2
- AB^2 = 49 + 784
- AB^2 = 833
- AB ≈ 28.87 cm
- In triangle ACD, using Pythagoras theorem, we can find the length of AD.
- AD^2 = CD^2 + AC^2
- AD^2 = 9^2 + 28^2
- AD^2 = 81 + 784
- AD^2 = 865
- AD ≈ 29.41 cm
- Now, in triangle ABD, using Pythagoras theorem, we can find the length of BD.
- BD^2 = AB^2 + AD^2
- BD^2 = 28.87^2 + 29.41^2
- BD^2 = 834.3369 + 866.3281
- BD^2 ≈ 1700.665
- BD ≈ 41.23 cm
Step 2: Find the area of quadrilateral ABCD
- Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ACD
- Area of triangle ABC = 1/2 * AB * BE
- Area of triangle ABC = 1/2 * 28.87 * 7
- Area of triangle ABC ≈ 101.045 cm^2
- Area of triangle ACD = 1/2 * AD * DF
- Area of triangle ACD = 1/2 * 29.41 * 9
- Area of triangle ACD ≈ 132.345 cm^2
- Area of quadrilateral ABCD = Area of triangle ABC + Area of triangle ACD
- Area of quadrilateral ABCD ≈ 101.045 + 132.345
- Area of quadrilateral ABCD ≈ 233.39 cm^2
Therefore, the area of quadrilateral ABCD is approximately 233.39 cm^2, which is closest to option B (224 cm^2).
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In a quadrilateral ABCD, the length of diagonal AC is 28 cm. The length of perpendicular from B and D on AC are 7 cm and 9 cm respectively. Then, the area of ABCD is:a)264b)224c)324d)214e)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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