Write an equation and solve. The longer leg of a right triangle is 7 cm more than the shorter le

step 1 okay this is a right angle triangle I have been given that the longer side is 7 inches more than

Write an equation and solve. The longer leg of a right...

Write an equation and solve.
The longer leg of a right triangle is $7 \mathrm{cm}$ more than the shorter leg. The length of the hypotenuse is $3 \mathrm{cm}$ more than twice the length of the shorter leg. Find the length of the hypotenuse.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in Euclidean geometry that establishes a relationship between the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, providing a method to calculate an unknown side length when the other two are known.

Variable Modeling in Geometry

Variable modeling in geometry involves assigning symbols to represent unknown quantities such as side lengths or angles. This allows the formulation of algebraic expressions and equations that capture the relationships described in a problem, enabling a systematic approach to finding solutions.

Algebraic Manipulation

Algebraic manipulation involves techniques such as expanding expressions, combining like terms, and isolating variables. In geometry problems, these skills are essential for translating geometric relationships into solvable algebraic equations and for simplifying those equations into a form where the unknown quantities can be determined.

Quadratic Equations

Quadratic equations are polynomial equations of degree two that often emerge when applying the Pythagorean Theorem or other geometric relationships that involve squared terms. Solving these equations may require techniques such as factoring, completing the square, or using the quadratic formula, all of which are central to obtaining the numerical solutions to problems involving squared relationships.

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