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XAT 2019 — QA & DI Question 16

Basics of TrianglesEasy

Let ABC be an isosceles triangle. Suppose that the sides AB and AC are equal and let the length of AB be x cm. Let b denote the angle ∠ABC and sin b = 3/5. If the area of the triangle ABC is M sq. cm, then which of the following is true about M?

Answer & solution

  • x2/4 ≤ M < x2/2

  • B

    3x2/4 ≤ M < x2

  • C

    M ≥ x2

  • D

    x2/2 ≤ M < 3x2/4

  • E

    M < x2/4

Solution

Now ∆ABC is an isosceles triangle where AB = AC. Let a perpendicular from A meet BC at D. As ∆ABC is isosceles, AD is a perpendicular bisector and BD = CD.

​​​​​​​

Given, Sin b = 3/5

⇒ Cos b = 4/5

In ∆ABD,

Sin b = 35 = ADAB

⇒ AD = 35AB = 3x5

Also, Cos b = 45 = BDAB

BD = 45AB = 4x5

∆ABC, BD = CD = 4x/5

∴ BC = 8x/5

⇒ Area of ∆ABC = ½ × BC × AD = ½ × 8x/5 × 3x/5 = 12x/25 = 0.48x.

Looking at the options we can see that M lies between x2/4 and x2/2. 

Hence, option (a).

XAT 2019 QA & DI Q16: Let ABC be an isosceles triangle. Suppose that the sides AB and AC are equal and let the length of AB be x cm. — Solution | TheCATExam