/*! 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 35 For Exercises 35 to \(38,\) make... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 35

For Exercises 35 to \(38,\) make drawings as needed. In parallelogram \(A B C D, A B=6 \mathrm{ft}\) and \(A D=10 \mathrm{ft}\). If \(m \angle A=65^{\circ}\) and \(\overline{B E}\) is the altitude to \(\overline{A D}\), find a) \(B E\) correct to tenths. b) the area of \(\square A B C D\)

Short Answer

Expert verified
a) \(BE \approx 9.1 \text{ ft}\); b) Area \(= 91 \text{ sq ft}\).

Step by step solution

01

Understand the Geometry

We have a parallelogram \(ABCD\), where \(AB = 6 \text{ ft}\) and \(AD = 10 \text{ ft}\). Angle \(A\) measures \(65^\circ\), and \(BE\) is an altitude from \(B\) to \(AD\).
02

Calculate Altitude BE

The altitude \(BE\) is the perpendicular distance from \(B\) to \(AD\). Using the relationship in right triangle \(ABE\), we apply the sine function: \(\sin(65^\circ) = \frac{BE}{AB} = \frac{BE}{10}\). Solving for \(BE\), \(BE = 10 \times \sin(65^\circ)\). Calculating gives \(BE \approx 9.1 \text{ ft}\).
03

Use Formula for Area of Parallelogram

The area of a parallelogram is given by the formula \(\text{Area} = \text{base} \times \text{height}\). Here, \(AD = 10 \text{ ft}\) is the base and \(BE \approx 9.1 \text{ ft}\) is the height. Thus, \(\text{Area} = 10 \times 9.1\).
04

Calculate Area

Multiply the base \(AD\) by the height \(BE\): \(\text{Area} = 10 \times 9.1 = 91 \text{ square feet}\).
05

Conclusion

The calculated altitude \(BE\) is approximately \(9.1 \text{ ft}\), and the area of parallelogram \(ABCD\) is \(91 \text{ square feet}\).

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.

Altitude
The altitude in a parallelogram is a critical concept in understanding its structure and area. It's essentially the height of the parallelogram, perpendicular from one vertex to the opposite side or the line extended from it. In our exercise, the altitude is represented by line segment \( BE \) from vertex \( B \) to line \( AD \). This specific line is crucial because it gives us the direct and shortest path from one side to the opposite, ensuring our area calculation is accurate.
  • In geometry terms, altitude is always perpendicular to the base it intersects or is drawn from.
  • Altitudes help in transforming complex quadrilaterals into more manageable segments, like right triangles, which can be solved using trigonometric functions like sine and cosine.
By calculating \( BE \) using sine (\( \sin 65^\circ = \frac{BE}{AD} \)), we find its value to be approximately \( 9.1 \) ft, essential for determining the parallelogram's area.
Angle Measurement
Measuring angles is foundational in solving many geometry problems, especially when dealing with shapes like parallelograms. An accurate reading allows us to apply trigonometry to find other necessary lengths and dimensions, as seen in this exercise.
  • In our example, angle \( \angle A = 65^\circ \) is the angle between sides \( AB \) and \( AD \).
  • The angle measurement establishes our basis for using trigonometric ratios to identify lengths, such as using \( \sin 65^\circ \).
When measuring angles, always note the vertices (angle notation) and use a protractor for accuracy if doing this manually. Here, the magnitude of \( \angle A \) directly enabled us to determine the sine ratio needed to compute the altitude.
Area Calculation
Calculating the area of geometric shapes, like parallelograms, enables us to understand their space coverage. For a parallelogram, the formula \( \text{Area} = \text{base} \times \text{height} \) holds true, just as it does for rectangles, though it's applied along one of its non-parallel sides.
  • In this exercise, \( AD = 10 \text{ ft} \) serves as the base, while our computed altitude \( BE = 9.1 \text{ ft} \) is the height.
  • Multiplying the base by the height gives us the total area of the parallelogram: \( 10 \times 9.1 = 91 \text{ square feet} \).
This clean, straightforward calculation using multiplication makes finding the area extremely feasible once both measurements are known.
Geometry Problem-Solving
Solving geometry problems involves a logical flow, use of appropriate formulas, and occasionally, drawing diagrams to visualize the problem. For a shape like a parallelogram, understanding its properties鈥攕uch as parallel opposite sides and equal opposite angles鈥攕treamlines the problem-solving process.
  • Begin with identifying all given values and symbols on a diagram.
  • Use these values to determine dependent measurements through trigonometry and geometry theorems.
In our problem, recognition of the right triangle within the parallelogram enabled the use of \( \sin 65^\circ \) to find the altitude. This blend of recognizing geometric properties, utilizing trigonometric functions, and systematic calculations are keys to mastering geometry problem-solving and unlocking solutions to complex spatial challenges.

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

In Exercises 27 to 34 , use the drawings where provided to solve each problem. Angle measures should be given to the nearest degree; distances should be given to the nearest tenth of a unit. Richard sees a balloon that is \(100 \mathrm{ft}\) above the ground. If the angle of elevation from Richard to the balloon is \(75^{\circ},\) how far from Richard is the balloon? (IMAGE CAN'T COPY)

Use a calculator and reciprocal relationships to find each ratio correct to four decimal places. $$\cot 67^{\circ}$$

Use the Law of Sines or the Law of cosines to solve each problem. Angle measures should be found to the nearest degree and areas and distances to the nearest tenth of a unit. A triangular lot has street dimensions of \(150 \mathrm{ft}\) and \(180 \mathrm{ft}\) and an included angle of \(80^{\circ}\) for these two sides. a) Find the length of the remaining side of the lot. b) Find the area of the lot in square feet.

In Exercises 27 to 34 , use the drawings where provided to solve each problem. Angle measures should be given to the nearest degree; distances should be given to the nearest tenth of a unit. A \(10-\) ft ladder is leaning against a vertical wall so that the bottom of the ladder is \(4 \mathrm{ft}\) away from the base of the wall. How large is the angle formed by the ladder and the wall?

Use the Law of Sines or the Law of cosines to solve each problem. Angle measures should be found to the nearest degree and areas and distances to the nearest tenth of a unit. Clay pigeons are released at an angle of \(30^{\circ}\) with the horizontal. Massimo hits one of the clay pigeons when shooting through an angle of elevation of \(70^{\circ} .\) If the point of release is \(120 \mathrm{m}\) from Massimo, how far \((x)\) is he from the target when it is hit?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.