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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.

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. Correct Answer 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

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