/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Find the area of each figure. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 11

Find the area of each figure. An isosceles right triangle with hypotenuse 8

Short Answer

Expert verified
The area of the isosceles right triangle is 16 square units.

Step by step solution

01

Calculate the length of the sides

To find the length of the two equal sides (Let's denote them as 'a'), we use the relationship between the hypotenuse (h) and the sides in an isosceles right triangle: \( a = \dfrac{h}{\sqrt{2}} \). Substituting the given value of h = 8, we can solve for a: \( a = \dfrac{8}{\sqrt{2}} = 8 \times \sqrt{2}/2 = 4 \times \sqrt{2} \)
02

Calculate the area of the triangle

The area (A) of a triangle is given by the formula \( A = \dfrac{1}{2} \times base \times height \). As this is an isosceles right triangle, the base and height are equal and can be replaced by a, therefore: \( A = \dfrac{1}{2} \times a \times a = \dfrac{1}{2} \times (4 \times \sqrt{2}) \times (4 \times \sqrt{2}) = 16 \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Isosceles Right Triangle
An isosceles right triangle is a special type of triangle. In this triangle, two sides are of equal length, and the angles opposite those sides are also equal. One angle is always 90 degrees, making it a right triangle. This unique combination forms a 45掳-45掳-90掳 triangle.
  • The equal sides are called the legs.
  • The side opposite the right angle is known as the hypotenuse.
  • These triangles always have a neat, symmetrical look.
When dealing with an isosceles right triangle, you can use these properties to easily figure out missing measurements.
Calculating the Area of a Triangle
The area of a triangle is a measure of the total space within its three sides. For any triangle, the area can be found using the formula: \( A = \dfrac{1}{2} \times \text{base} \times \text{height} \). In an isosceles right triangle, both the base and height are the same, which simplifies the calculation.
  • Set the base and height equal to the length of the legs, which we found using the hypotenuse.
  • Apply the formula: \( A = \dfrac{1}{2} \times a \times a \), where \( a \) is the length of the legs.
This results in an efficient way to calculate the area using only the known hypotenuse.
Importance of the Hypotenuse
The hypotenuse is the longest side in a right triangle, opposite the right angle. In our case of the isosceles right triangle:
  • You can use the Pythagorean theorem, but a shortcut exists for these specific triangles.
  • The relationship \( a = \dfrac{h}{\sqrt{2}} \) helps us find the leg length, where \( h \) is the hypotenuse.
Knowing the hypotenuse simplifies calculating other properties of the triangle, proving its vital role in geometry.
Exploring Triangle Properties
Triangle properties lay the foundation for solving and understanding mathematical problems about triangles. Isosceles right triangles have patterns and symmetry that make finding missing values straightforward:
  • Equal leg lengths create a balance that simplifies many calculations.
  • The 45-degree angles show the internal symmetry.
  • All properties of an isosceles right triangle can be utilized to solve for areas, perimeters, and more.
These properties are not only fascinating but also extremely helpful in geometry, making them worth understanding and remembering.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If 6 oz of dough are needed to make an 8 -in. pizza, how much dough will be needed to make a 16 -in. pizza of the same thickness? (Hint: Compare the areas of the pizza tops.)

Find the area of each trapezoid. An isosceles trapezoid with legs 13 and bases 10 and 20.

If a value of \(\pi\) is required in the following exercises, use \(\pi \approx 3.14\) Darts are thrown at a 1 -meter square which contains an irregular red region. Of 100 darts thrown, 80 hit the square. Of these, 10 hit the red region. Estimate the area of this region.

Find the area of each polygon. A regular hexagon with perimeter 72

Complete the table. Leave answers in terms of \(\pi\). $$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline & 1 . & 2 . & 3 . & 4 . & 5 . & 6 . & 7 . & 8 . \\\\\hline \text { Radius } & 7 & 120 & \frac{5}{2} & 6 \sqrt{2} & ? & ? & ? & ? \\\\\hline \text { Circumference } & ? & ? & ? & ? & 20 \pi & 12 \pi & ? & ? \\\\\hline \text { Area } & ? & ? & ? & ? & ? & ? & 25 \pi & 50 \pi \\\\\hline\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.