Solving a Right-Angled Triangle Using Algebra and the Pythagorean Theorem

In mathematics, particularly in geometry, solving right-angled triangles involves applying both algebra and the Pythagorean theorem. This article guides you through a step-by-step process to solve a specific right-angled triangle where the hypotenuse, other side, and shortest side have specific relationships to each other. The main goal is to determine the lengths of all three sides of the triangle.

Introduction to the Problem

Consider a right-angled triangle where the hypotenuse is 4 cm more than the shortest side, and the third side is 2 cm less than the hypotenuse. The problem can be represented by the following relationships:

The hypotenuse is 4 cm more than the shortest side: The third side is 2 cm less than the hypotenuse.

Setting Up the Variables and Equations

To solve this problem, we denote the sides of the right triangle as follows:

a the shortest side b the other side c the hypotenuse

Given the relationships:

c a 4 b c - 2

By substituting the expression for c from the first equation into the second equation:

b (a 4) - 2 a 2

Thus, we have:

Sides: Shortest side (a), Other side (b a 2), Hypotenuse (c a 4)

Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore:

a^2 b^2 c^2

Substituting the expressions for b and c into this equation:

a^2 (a 2)^2 (a 4)^2

Expanding the squares:

a^2 a^2 4a 4 a^2 8a 16

Combining like terms:

2a^2 4a 4 - a^2 - 8a - 16 0

Which simplifies to:

a^2 - 4a - 12 0

Factoring the quadratic equation:

(a - 6)(a 2) 0

Setting each factor to zero gives us:

a - 6 0 or a 2 0

Solving for a gives us:

a 6 (rejecting the negative root since side lengths cannot be negative)

Calculating the Sides of the Triangle

Now that we have determined a 6 cm, we can find b and c:

b a 2 6 2 8 cm c a 4 6 4 10 cm

Therefore, the sides of the triangle are:

Shortest side (a) 6 cm Other side (b) 8 cm Hypotenuse (c) 10 cm

Summary of the Triangle Sides