Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch?

Step-by-Step Solution

Answer

The two equal sides of the isosceles right triangle are of length \( sqrt(2) \) inches each.

Step 1: Understand the Structure of the Triangle

An isosceles right triangle is a specific type of right triangle where the two legs are of equal length. Let's call the length of one of the legs 'a'. Therefore, in this case, the two legs are a and a, and the hypotenuse is 2 inches.

Step 2: Use the Pythagorean Theorem

The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. We could denote this as \( a^2 + a^2 = 2^2 \). Solving for 'a' will give us the length of the two equal sides.

Step 3: Solve for a

Adding up \( a^2 + a^2 \) gives \( 2a^2 \). Setting this equal to \(2^2\) or 4, we find \(2a^2 = 4\). Solving for 'a', we get \(a = sqrt(2)\) inches.

Step 4: Double Check with Area

Now, we double check our solution with the given area of the triangle. The formula for the area of a triangle is 1/2 * base * height. As this is an isosceles right triangle, base and height are equal, in our case both are a. Therefore, the area is \(1/2 * sqrt(2) * sqrt(2)\), which equals to 1 square inch confirming that our solution is correct.

Key Concepts

Pythagorean Theorem

When dealing with isosceles right triangles, the Pythagorean Theorem is incredibly handy. This theorem helps you find the relationship between the sides of a right triangle. In simple terms, it tells us that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. For our triangle with sides labeled as 'a' and a hypotenuse of 2 inches, we write this as:\[ a^2 + a^2 = 2^2 \]This equation simplifies our search for the missing sides. So, whenever you're dealing with a right triangle, always remember the Pythagorean Theorem. It's a pillar of triangle mathematics that aids us in solving various geometrical problems.

Triangle Area Formula

The area formula for a triangle is essential in confirming the dimensions of our triangle. For any triangle, the area can be calculated using the formula:

• \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)

In an isosceles right triangle, the base and the height are equal. This exercise states an area of 1 square inch, so we set up our formula to confirm our dimensions:\[ \text{Area} = \frac{1}{2} \times a \times a = 1 \]Calculating with \( a = \sqrt{2} \), verifies the area matches the given data. These steps are crucial to ensuring your results align with the triangle's supposed characteristics.

Hypotenuse

The hypotenuse in a right triangle is the side opposite the right angle, and it's always the longest side. In this problem, you're given a hypotenuse of 2 inches. The hypotenuse is critical because it helps you leverage the Pythagorean Theorem to find the lengths of the other sides. Once you learn the hypotenuse's length, you effectively have a key to unlock the triangle's full dimensions. It's the side that bridges the other two smaller sides and plays a crucial role in solving the triangle.

Equal Sides

A unique feature of an isosceles right triangle is its two equal sides, called the legs. These equal sides make this triangle particularly interesting and give it symmetry. In mathematical expressions, we denote these equal sides in our problem as 'a'.Understanding that the sides are equal simplifies many calculations since it reduces the number of unknowns you are solving for. When you use the Pythagorean Theorem, knowing the sides are equal means you only need to solve for one variable (a) to find both sides.\[ 2a^2 = 4 \Rightarrow a^2 = 2 \Rightarrow a = \sqrt{2} \]Having this knowledge assists greatly in quickly solving and confirming the validity of your results for isosceles right triangles.