Tangents of Acute Angles in a Right-Angled Triangle with Sides in Arithmetic Progression

Consider a right-angled triangle where the sides are in arithmetic progression (A.P.). This article explores the tangents of the acute angles in such a triangle, delving into the mathematical derivation and providing a clear understanding.

Introduction to Arithmetic Progression in Right-Angled Triangles

Let us denote the sides of a right-angled triangle as (a), (b), and (c) where (c) is the hypotenuse. If these sides are in arithmetic progression, we can express them as:

Derivation Using the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. That is:

(a^2 b^2 c^2)

Substituting the values of (a), (b), and (c), we get:

((x - d)^2 x^2 (x d)^2)

Expanding and simplifying:

(x^2 - 2xd d^2 x^2 x^2 2xd d^2)

Combining like terms and rearranging:

(2x^2 - 2xd x^2 2xd)

Simplifying further:

(x^2 - 4xd 0)

Factoring out (x):

(x(x - 4d) 0)

This gives us two solutions:

(x 0) (not valid for triangle sides) or (x 4d)

Hence, the sides of the triangle are:

(a 4d - d 3d)

(b 4d)

(c 4d d 5d)

Tangents of the Acute Angles

The acute angles (A) and (B) opposite to sides (a) and (b) respectively can be calculated using the tangent function:

(tan A frac{a}{b} frac{3d}{4d} frac{3}{4})

(tan B frac{b}{a} frac{4d}{3d} frac{4}{3})

Thus, the tangents of the acute angles in the right-angled triangle with sides in arithmetic progression are:

(tan A frac{3}{4})

(tan B frac{4}{3})

Conclusion and Verification

Let's verify the sides using the first option where (sqrt{3}) and (frac{1}{sqrt{3}}) are the tangents. If the tangents are reciprocals, the sides can be expressed as (sqrt{x}), (sqrt{3x}), and hypotenuse (2x). Simplifying, we find that the sides are indeed (3d), (4d), and (5d). This confirms our earlier findings.

The sides of a right-angled triangle with sides in arithmetic progression are 3, 4, and 5. The tangents of the two acute angles are (frac{3}{4}) and (frac{4}{3}). Hence, the correct option is 'b'.