Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, ACB = 90° and AC = 15 cm.
Answers
Given : In quadrilateral ABCD , AB = 17 cm, AD = 9 cm, CD = 12 cm, ACB = 90° and AC = 15 cm.
To find : Perimeter and area of the quadrilateral ABCD.
By using Pythagoras theorem in right ∆ACB :
AB² = BC² + AC²
17² = BC² + 15²
BC² =17² - 15²
BC² = 289 - 225
BC² = 64
BC = √64
BC = 8 cm
Perimeter of quadrilateral ABCD , P = AB + BC + CD + DA
P = 17 + 9 + 12 + 8
P = 46 cm
Perimeter of quadrilateral ABCD = 46 cm
Area of right triangle △ACB, A = ½ × base × height
A = 12 × 8 x 15
Area of right triangle △ACB = 60 cm²
Now, for the area of the ∆ ACD :
Let a = 15 cm, b = 12 cm and c = 9 cm
Semi perimeter of Δ (s) = (a + b + c)/2
s = (15 + 12 + 9)/2
s = 36/2
s = 18 cm
By using Heron's formula :
Area of ∆ ACD, A = s√(s − a)(s − b)(s − c)
A = 18√(18 - 15)(18 - 12)(18 - 9)
A = √18 × (3 × 6) × 9
A =√(18 × 18) × (3 × 3)
A = 18 × 3
A = 54 cm²
Area of ∆ ACD = 54 cm²
Area of quadrilateral ABCD= Area of △ACB + Area of △ACD
Area of quadrilateral ABCD = 60 + 54 = 114 cm²
Hence, the perimeter of quadrilateral ABCD is 46 cm and the area of quadrilateral ABCD is 114 cm².
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Answer:
To find : Perimeter and area of the quadrilateral ABCD.
By using Pythagoras theorem in right ∆ACB : AB² = BC² + AC²
17² = BC² + 15²
BC² =17² - 15²
BC² = 289 - 225
BC² = 64 BC = √64 BC = 8 cm
Perimeter of quadrilateral ABCD , P = AB + BC + CD + DA P = 17 + 9 + 12 + 8 P = 46 cm
Perimeter of quadrilateral ABCD = 46 cm
Area of right triangle △ACB, A = ½ × base × height
A = 12 × 8 x 15
Area of right triangle △ACB = 60 cm²
Now, for the area of the ∆ ACD : Let a = 15 cm, b = 12 cm and c = 9 cm
Semi perimeter of Δ (s) = (a + b + c)/2
s = (15 + 12 + 9)/2
s = 36/2
s = 18 cm
By using Heron's formula : Area of ∆ ACD, A = s√(s − a)(s − b)(s − c) A = 18√(18 - 15)(18 - 12)(18 - 9)
A = √18 × (3 × 6) × 9
A =√(18 × 18) × (3 × 3)
A = 18 × 3
A = 54 cm
total area = 60+54=114cm
finally, perimeter=46cm
area=114cm