Question : What is the area of a triangle having a perimeter of 32 cm, one side of 11 cm, and the difference between the other two sides is 5 cm?
Option 1: $8\sqrt{30}$ cm2
Option 2: $5\sqrt{35}$ cm2
Option 3: $6\sqrt{30}$ cm2
Option 4: $8\sqrt{2}$ cm2
Correct Answer: $8\sqrt{30}$ cm2
Solution : Let the sides of the triangle be $a$, $b$, and $c$. Given Perimeter = 32 cm ⇒ $a+b+c=32$ ...... equation (1) One side, let $a$ =11 cm Also, the difference between the other two sides, $b−c=5$........ equation (2) Performing equation (1) + equation (2) $a+2b=37$ ⇒ $11+2b=37$ ⇒ $2b=26$ ⇒ $b=13$ cm Now, using equation (2), $c=b−5$ ⇒ $c=13−5=8$ cm Area of the triangle according to Heron's formula = $\sqrt{s(s−a)(s−b)(s−c)}$ where $s$ is the semi-perimeter $s=\frac{11+13+8}{2} = 16$ cm Area $= \sqrt{16(16−11)(16−13)(16−8)}$ $= \sqrt{16(5)(3)(8)}$ $= 8\sqrt{30}$ cm2 Hence, the correct answer is $8\sqrt{30}$ cm2.
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Question : The difference between the semi-perimeter and the sides of ΔPQR are 18 cm, 17 cm, and 25 cm, respectively. Find the area of the triangle.
Option 1: $330\sqrt{510}$ cm2
Option 2: $230\sqrt{510}$ cm2
Option 3: $30\sqrt{510}$ cm2
Option 4: $130\sqrt{510}$ cm2
Question : The base of a triangle is equal to the perimeter of a square whose diagonal is $6 \sqrt{2}$ cm, and its height is equal to the side of a square whose area is 144 cm2. The area of the triangle (in cm2) is:
Option 1: 288
Option 2: 216
Option 3: 144
Option 4: 72
Question : The sides of a triangle are 8 cm, 12 cm, and 16 cm. What is the area of the triangle?
Option 1: $24 \sqrt{15}~\text{cm}^2$
Option 2: $6 \sqrt{15}~\text{cm}^2$
Option 3: $8 \sqrt{15}~\text{cm}^2$
Option 4: $12 \sqrt{15}~\text{cm}^2$
Question : The base of a triangle is equal to the perimeter of a square whose diagonal is $9 \sqrt{2}$ cm and its height is equal to the side of a square whose area is 144 cm2. The area of the triangle(in cm2) is:
Option 1: 216
Option 2: 72
Option 4: 288
Question : The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 90 cm. Find its area (in cm2).
Option 1: 150
Option 2: 270
Option 3: 30
Option 4: 60
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