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Animated Solution for Mathematics - Sequence and Series: 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 ?

Q: Why did we choose $a-d, a, a+d$ instead of $a, a+d, a+2d$?

Great question! While both work, using $a-d, a, a+d$ is a classic algebraic trick. It centers the variables around $a$, which leads to beautiful cancellations in the expansion of $(a-d)^2$ and $(a+d)^2$. It makes the math much cleaner and less prone to errors!

Q: Why do we reject the negative root for $d$?

In geometry, side lengths must be positive. If $d$ were negative, our side $a-d$ would become $a + |d|$, and our side $a+d$ would become $a - |d|$. While mathematically valid in some contexts, for a physical triangle, we define $d$ as the common difference such that the sides remain positive. Here, $d=2$ gives us sides $6, 8, 10$, which are all positive.

Enter Numerical Value:

Visualized Solution

The Right-Angled Triangle

Given: Right-angled triangle
Sides are in Arithmetic Progression (A.P.)
Area = $24$

Assuming Sides in A.P.

Let the sides be $a - d$ , $a$ , and $a + d$ (where $d > 0$ )
The hypotenuse must be the longest side: $a + d$

Applying Pythagoras Theorem

Apply Pythagoras Theorem:
$(Base)^{2} + (Perpendicular)^{2} = (Hypotenuse)^{2}$

Substituting the Sides

Substitute the assumed sides into the theorem:
$(a - d)^{2} + a^{2} = (a + d)^{2}$

Expanding the Squares

Expand using $(x \pm y)^{2} = x^{2} \pm 2 x y + y^{2}$ :
$(a^{2} - 2 a d + d^{2}) + a^{2} = a^{2} + 2 a d + d^{2}$

Simplifying the Equation

Cancel common terms ( $a^{2}$ and $d^{2}$ ) on both sides:
$a^{2} - 2 a d = 2 a d$
$a^{2} = 4 a d$

Finding the Relation

Since $a$ is a side length, $a \neq = 0$ .
Divide both sides by $a$ :
$a = 4 d$

Sides in Terms of $d$

Substitute $a = 4 d$ into the sides:
Perpendicular: $a - d = 4 d - d = 3 d$
Base: $a = 4 d$
Hypotenuse: $a + d = 4 d + d = 5 d$

Area of the Triangle

Formula for Area of a Right-Angled Triangle:
$Area = \frac{1}{2} \times Base \times Perpendicular$

Substituting Sides into Area

Substitute the base ( $4 d$ ) and perpendicular ( $3 d$ ):
$Area = \frac{1}{2} \times (4 d) \times (3 d)$

Simplifying the Area

Simplify the expression:
$Area = \frac{12 d ^{2}}{2}$
$Area = 6 d^{2}$

Equating to Given Area

We are given that the Area = $24$
Equate the expressions:
$6 d^{2} = 24$

Solving for $d$

Divide by $6$ :
$d^{2} = 4$
Take the square root:
$d = 2$ (since $d > 0$ )

Finding the Smallest Side

Identify the smallest side: $3 d$
Substitute $d = 2$ :
$Smallest side = 3 (2) = 6$

Conclusion

Key Takeaway:
Right-angled triangle sides in A.P. are always in the ratio $3 : 4 : 5$ .
The sides are $6, 8, 10$ .

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The Sigma Insight: Arithmetic Progression (A.P.)

Solution Diagram

Analyzing the Setup

To solve this problem, we represent the sides of the right-angled triangle as an Arithmetic Progression (A.P.). Let the sides be

a - d

a

, and

a + d

, where

d > 0

This symmetric choice of variables simplifies the algebraic expansion. Since the hypotenuse is always the longest side in a right-angled triangle, we identify the hypotenuse as

a + d

The Pythagorean Bridge

We apply the Pythagorean theorem, which states that the sum of the squares of the two shorter sides equals the square of the hypotenuse:

(a - d)^{2} + a^{2} = (a + d)^{2}

Expanding both sides of the equation, we obtain:

(a^{2} - 2 a d + d^{2}) + a^{2} = a^{2} + 2 a d + d^{2}

Simplifying the expression by canceling common terms, we arrive at:

a^{2} - 2 a d = 2 a d

a^{2} = 4 a d

Since

a

represents a side length, $a eq 0$. Dividing by

a

, we find the fundamental relationship:

a = 4 d

The Ratio Revealed

Substituting

a = 4 d

back into our side expressions, the sides of the triangle become

3 d

4 d

, and

5 d

. This confirms that any right-angled triangle with sides in A.P. must follow the $3 : 4 : 5$ ratio.

The area of a right-angled triangle is given by the formula:

Area = \frac{1}{2} \times base \times height

Substituting our sides into the area formula:

Area = \frac{1}{2} \times (3 d) \times (4 d) = 6 d^{2}

Given that the area is

24

, we set up the equation:

6 d^{2} = 24

d^{2} = 4

d = 2

Final Calculation

The smallest side of the triangle is defined as

3 d

. Substituting the value of

d = 2

Smallest Side = 3 \times 2 = 6

The sides of the triangle are

6

8

, and

10

. Thus, the smallest side is $6$ .

Common Traps & FAQs

Why did we choose

a - d, a, a + d

instead of

a, a + d, a + 2 d

Great question! While both work, using

a - d, a, a + d

is a classic algebraic trick. It centers the variables around

a

, which leads to beautiful cancellations in the expansion of

(a - d)^{2}

and

(a + d)^{2}

. It makes the math much cleaner and less prone to errors!

Why is the hypotenuse always

a + d

In any right-angled triangle, the hypotenuse is the longest side. Since

d

must be positive (otherwise the sides would be negative or zero, which is impossible for a triangle),

a + d

is clearly the largest of the three terms

a - d

a

, and

a + d

Why do we reject the negative root for

d

In geometry, side lengths must be positive. If

d

were negative, our side

a - d

would become

a + ∣ d ∣

, and our side

a + d

would become

a - ∣ d ∣

. While mathematically valid in some contexts, for a physical triangle, we define

d

as the common difference such that the sides remain positive. Here,

d = 2

gives us sides

6, 8, 10

, which are all positive.

Animated Solution for Mathematics - Sequence and Series: 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 ?

Enter Numerical Value:

Visualized Solution

The Right-Angled Triangle

Assuming Sides in A.P.

Applying Pythagoras Theorem

Substituting the Sides

Expanding the Squares

Simplifying the Equation

Finding the Relation

Sides in Terms of $d$

Area of the Triangle

Substituting Sides into Area

Simplifying the Area

Equating to Given Area

Solving for $d$

Finding the Smallest Side

Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

The Pythagorean Bridge

The Ratio Revealed

Final Calculation

Similar Questions

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ} .$ If the largest interior angle of the polygon is $21 9^{\circ}$ , then n is equal to

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is :

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $a_{1} + a_{2} + a_{3} + a_{4} = 48$ and $a_{1} a_{2} a_{3} a_{4} + l^{4} = 361$ , then the largest term of the A.P. is equal to

If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6 : 11$ and the seventh term lies in between $130$ and $140$ , then the common difference of this A.P. is \dots.

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac{1}{2}$ , then the greatest number amongst them is

Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. If $a_{7} = 3$ , the product $a_{1} a_{4}$ is minimum and the sum of its first $n$ terms is zero, then $n! - 4 a_{n (n + 2)}$ is equal to :

If $lo g_{3} 2, lo g_{3} (2^{x} - 5), lo g_{3} (2^{x} - \frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to

Animated Solution for Mathematics - Sequence and Series: 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 ?

Enter Numerical Value:

Visualized Solution

The Right-Angled Triangle

Assuming Sides in A.P.

Applying Pythagoras Theorem

Substituting the Sides

Expanding the Squares

Simplifying the Equation

Finding the Relation

Sides in Terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Area of the Triangle

Substituting Sides into Area

Simplifying the Area

Equating to Given Area

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Finding the Smallest Side

Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

The Pythagorean Bridge

The Ratio Revealed

Final Calculation

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The interior angles of a polygon with n sides, are in an A.P. with common difference 6∘. If the largest interior angle of the polygon is 219∘, then n is equal to

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The fourth power of the common difference of an arithmetic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a1​,a2​,a3​,… be an A.P. If a7​=3, the product a1​a4​ is minimum and the sum of its first n terms is zero, then n!−4an(n+2)​ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If log3​2,log3​(2x−5),log3​(2x−27​) are in an arithmetic progression, then the value of x is equal to

Sides in Terms of $d$

Solving for $d$

The interior angles of a polygon are in arithmetic progression. The smallest angle is $12 0^{\circ}$ , and the common difference is $5^{\circ}$ , Find the number of sides of the polygon.

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ} .$ If the largest interior angle of the polygon is $21 9^{\circ}$ , then n is equal to

If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac{1}{2}$ , then the greatest number amongst them is

Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. If $a_{7} = 3$ , the product $a_{1} a_{4}$ is minimum and the sum of its first $n$ terms is zero, then $n! - 4 a_{n (n + 2)}$ is equal to :

If $lo g_{3} 2, lo g_{3} (2^{x} - 5), lo g_{3} (2^{x} - \frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to