The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side? [2018]

Question

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side? [2018]

Smart Achievers · Accepted Answer

(6)

Let the sides be $a - d, a, a + d$ where $d$ is positive. Using Pythagoras theorem,

${(a + d)}^{2} = {(a - d)}^{2} + a^{2} \Rightarrow a = 4 d$

$∴$ $Sides are 3 d, 4 d, 5 d$

$Area = 24 \Rightarrow \frac{1}{2} \times 3 d \times 4 d = 24 \Rightarrow d^{2} = 4 \Rightarrow d = 2$

$∴$ $Smallest side = 3 d = 6 .$

Solution

Q.
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side [2018]

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Solution

Q. The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side [2018]

Ans. (6) Let the sides be a-d, a, a+d where d is positive. Using Pythagoras theorem, (a+d)2=(a-d)2+a2⇒a=4d ∴ Sides are 3d, 4d, 5d Area=24⇒12×3d×4d=24⇒d2=4⇒d=2 ∴ Smallest side =3d=6.

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Q.
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side [2018]