Problem 46 - Algebra 2 and Trigonometry

Question

In \(43-47,\) express each answer in simplest radical form. The length of each leg of an isosceles right triangle is \(\sqrt{98}\) inches. a. Find the length of the hypotenuse. b. What is the perimeter of the triangle?

Step-by-Step Solution

Answer

Hypotenuse: 14 inches; Perimeter: \(14\sqrt{2} + 14\) inches.

Step 1: Identify Triangle Properties

An isosceles right triangle means that two sides are equal, and the non-hypotenuse angles are 45 degrees each. The hypotenuse can be found using the Pythagorean theorem since both legs (sides) of the triangle are equal.

Step 2: Apply the Pythagorean Theorem

Use the Pythagorean theorem \(a^2 + a^2 = c^2\) for an isosceles right triangle. Substituting \(a = \sqrt{98}\), we get \(2(\sqrt{98})^2 = c^2\).

Step 3: Simplify and Solve for Hypotenuse

Calculate \((\sqrt{98})^2 = 98\), so \(2 \times 98 = c^2\). Thus, \(c^2 = 196\). Solve for \(c\): \(c = \sqrt{196} = 14\). The length of the hypotenuse is 14 inches.

Step 4: Finding the Perimeter of the Triangle

The perimeter is the sum of the lengths of all sides. Since the lengths of the legs are \(\sqrt{98}\) and the hypotenuse is 14, the perimeter is \(\sqrt{98} + \sqrt{98} + 14 = 2\sqrt{98} + 14\).

Step 5: Simplify the Perimeter

Simplify \(2\sqrt{98}\) by expressing \(\sqrt{98}\) as \(\sqrt{49 \times 2} = 7\sqrt{2}\). So, \(2\sqrt{98} = 2 \times 7\sqrt{2} = 14\sqrt{2}\). The perimeter is \(14\sqrt{2} + 14\).

Key Concepts

Isosceles Right Triangle

An isosceles right triangle is a type of triangle with two equal sides and a right angle. These triangles have unique properties that make them special. If you look at an isosceles triangle, you'll notice that besides the two equal sides, there's a 90-degree angle opposite the hypotenuse, which is the longest side. Furthermore, the angles opposite the equal sides both measure 45 degrees each.

This special configuration is why sometimes people call them 45-45-90 triangles. These characteristics are essential, especially for calculations involving the Pythagorean Theorem, as you'll learn next.

Simplest Radical Form

The term 'simplest radical form' refers to expressing a number in a simplified way when it involves square roots. Let's consider the square root of 98, which was given as the length of the sides in the exercise.

To simplify \( \sqrt{98} \):

• Look for a perfect square factor of 98. In this case, 49 is a perfect square, as \( 49 \times 2 = 98 \).
• Thus, \( \sqrt{98} = \sqrt{49 \times 2} \).
• You can simplify it further into \( 7\sqrt{2} \) because \( \sqrt{49} = 7 \).

This process ensures we express the radical in its simplest form, making further calculations more manageable and less prone to errors.

Perimeter Calculation

Calculating the perimeter of an isosceles right triangle involves adding together the lengths of all three sides. For our specific triangle, the two legs are equal lengths of \( \sqrt{98} \), and the hypotenuse is 14 inches, as determined previously using the Pythagorean Theorem.

Let's put it together:

• The formula is \( \text{Perimeter} = a + a + c \), where \( a = \sqrt{98} \) and \( c = 14 \).
• This becomes \( \sqrt{98} + \sqrt{98} + 14 \).
• We already determined \( \sqrt{98} \) in its simplest form as \( 7\sqrt{2} \). Therefore, \( 2\sqrt{98} = 2 \times 7\sqrt{2} = 14\sqrt{2} \).

So, the perimeter in simplest form is \( 14\sqrt{2} + 14 \). This hybrid expression involving a constant term and a radical term illustrates how elements blend together, giving the total distance around our triangle.