The following questions have three statements. Study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. Find the area of the isosceles triangle. I: The length of median to the side opposite to the single largest angle in the triangle is 3√5 cm. II: The length of the side opposite to the single largest angle in the triangle is 12 cm. III: The perimeter of the triangle is 30 cm.?

Option 2 : Any two statements are sufficient to answer the question.

Statement I and II:

The length of median to the side opposite to the single largest angle in the triangle is 3√5 cm and the length of the side opposite to the single largest angle in the triangle is 12 cm;

Side opposite to the single largest angle will be longest.

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From the figure:

We can find the value of AB and AC using Pythagoras theorem.

∴The area of the triangle can be find out with help of lengths of all three sides.

Statement II and III:

The length of the side opposite to the single largest angle in the triangle is 12 cm and perimeter of the triangle is 30 cm;

∴ The equal sides of the triangle = (30 – 12)/2 = 9 cm

∴ The area of the triangle can be find out with help of lengths of all three sides.

Statement I and III:

AB + BC + AC = 30

⇒ 2AB + BC = 30

⇒ AB = (30 – BC)/2      ----(1)

Using Apollonius’ theorem

AB² + AC² = 2CD² + 2AD²

⇒ 2AB² = 2CD² + 2AD²      (∵ ΔABC is isosceles triangle)

⇒ AB² – (BC/2)² = AD²      (∵ 2CD = BC)

⇒ (30 – BC)² – BC² = 4(3√5)²      {from eq. (1)}

⇒ 900 + BC² – 60BC – BC² = 180

⇒ BC = 12 cm

Area of triangle = 1/2 × BC × AD = 1/2 × 12 × 3√5

∴ Any two of the statements are sufficient to answer the question