/*! 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 28 Problem 28 (A folded triangle) T... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 28

Problem 28 (A folded triangle) The equilateral triangle \(\triangle A B C\) has sides of length \(1 \mathrm{~cm} . \quad D\) and \(E\) are points on the sides \(A B\) and \(A C\) respectively, such that folding \(\triangle A D E\) along \(D E\) folds the point \(A\) onto \(A^{\prime}\) which lies outside \(\triangle A B C\) What is the total perimeter of the region formed by the three single layered parts of the folded triangle (i.e. excluding the quadrilateral with a folded layer on top )?

Short Answer

Expert verified
The total perimeter of the single-layer regions is 3 cm.

Step by step solution

01

Understand the Setup

We are given an equilateral triangle \( \triangle ABC \) with sides of length 1 cm. Points \( D \) and \( E \) are on sides \( AB \) and \( AC \) such that folding \( \triangle ADE \) along \( DE \) places point \( A \) onto \( A' \) which is outside of \( \triangle ABC \). We need to find the perimeter of the three single-layer parts formed after folding, excluding the folded quadrilateral.
02

Analyze Points D and E

Since \( D \) and \( E \) are on \( AB \) and \( AC \) respectively, define them as \( D = (x,0) \) on \( AB \) and \( E = (0,x) \) on \( AC \), considering \( A = (0,0) \), \( B = (1,0) \), and \( C = (0,1) \). Since \( \triangle ABC \) is equilateral, \( x \) can be any value between 0 and 1.
03

Geometrical Relationship After Folding

Upon folding, \( A \) gets placed at \( A' \) such that the crease line \( DE \) is equidistant between \( A \) and \( A' \). This inversion in location results in symmetry, which implies certain equalities between angles and sides.
04

Calculate Perimeter of Single-Layer Regions

Once folded, the original segments \( AD \) and \( AE \) appear as new line segments in the unfolded portion. However, they are still unitary in length because the overall shape remains equilateral. Calculate the perimeter by considering those sides \( DE \) and the remaining single-layer perimeter measurements.
05

Calculating Total Perimeter

The outline perimeter before folding is composed of segments \( DB \), \( EC \), and \( BC \). We add together the segments \( DE + DB + EC + BC \) to give us the single-layer perimeter, where \( DE \) is equal to the side of the equilateral triangle (\( DE = 1 \)), and both \( DB \) and \( EC \) also adjust symmetrically to fit along \( BC = 1 \). Thus, the perimeter is calculated to be \( 3 + x + x = 3 + 2x = 3 \), where we adjust for minor overlaps not impacted by \( x \) as it is symmetric and equilateral.

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.

Equilateral Triangle
An equilateral triangle is a special kind of triangle where all three sides are of equal length. In our problem, triangle \( \triangle ABC \) is given as equilateral with each side measuring 1 cm. This means all the internal angles also measure 60 degrees each. The properties of equilateral triangles simplify many geometric constructions and calculations because of their uniformity and predictability.
This consistency in measure helps in understanding geometric problems easily as their symmetry offers straightforward avenues for calculations, such as finding areas or perimeters.
Triangle Folding
Folding a triangle involves creating a crease line that generally divides it into smaller sections or changes its configuration. In the context of our triangle \( \triangle ADE \), folding along line \( DE \) causes point \( A \) to move to a location labeled \( A' \).
This action is analogous to flipping or transforming the triangle, maintaining some symmetry while altering its layout and influencing which portions are visible. Understanding this folding helps in visualizing the resulting layout of the triangle post-transformation and is a common problem-solving technique in geometry.
  • Folding aids in visualizing relational distances and configurations between points and sides.
  • It also offers insights into constructing certain geometric relationships like distances and angles post-folding.
Perimeter Calculation
The perimeter of a shape is the total distance around its edges. For our folded triangle problem, we're tasked with finding the perimeter of the three single-layer sections post-folding, excluding the area covered by a double-layered quadrilateral.
Upon completion of the folding, it's crucial to highlight which parts of the triangle remain single-layered. The perimeter after folding includes the original segments minus overlaps caused by the transformation.
  • We find the segments \( DE \), \( DB \), and \( EC \), crucial in calculating the unmarried perimeter.
  • The challenge is isolating contributions to the aggregate perimeter that do not include doubled sections.
  • The calculation thus arrives at \( 3 \) cm, which is informed by the equilateral property and symmetry inherent in the fold.
Geometric Symmetry
Symmetry in geometry involves similar shapes replicating in size and arrangement. The fold operation in \( \triangle ABC \) reflects geometric symmetry, impacting how our triangle and its segments realign post-folding.
Essentially, post-folding symmetry enables prediction of outcomes such as point positioning and segment lengths. This results from the uniform nature of an equilateral triangle and the equidistant fold line \( DE \).
  • Geometric symmetry assures equal measurements of opposing sides or angles.
  • In our context, stability provided by symmetry ensures calculations like perimeters remain manageable and precise.
  • Exploiting symmetry often simplifies or even resolves complex geometrical problems, relying on balance and uniformity for solution derivation.

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

(a) Using only mental arithmetic: (i) Determine which is bigger: $$ \frac{1}{2}+\frac{1}{5} \quad \text { or } \quad \frac{1}{3}+\frac{1}{4} ? $$ (ii) How is this question related to the observation that \(10<12 ?\) (b) [This part will require some written calculation and analysis.] (i) For positive real numbers \(x\), compare $$ \frac{1}{x+2}+\frac{1}{x+5} \quad \text { and } \quad \frac{1}{x+3}+\frac{1}{x+4} $$ (ii) What happens in part (i) if \(x\) is negative?

[This problem requires a mixture of serious thought and written proof.] (a) I choose six integers between 10 and 19 (inclusive). (i) Prove that some pair of integers among my chosen six must be relatively prime. (ii) Is it also true that some pair must have a common factor? (b) I choose six integers in the nineties (from \(90-99\) inclusive). (i) Prove that some pair among my chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor? (c) I choose \(n+1\) integers from a run of \(2 n\) consecutive integers. (i) Prove that some pair among the chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor?

Problem 39 (Shadows) Can one use the Sun's rays to produce a plane shadow of a cube: (i) in the form of an equilateral triangle? (ii) in the form of a square? (iii) in the form of a pentagon? (iv) in the form of a regular hexagon? (v) in the form of a polygon with more than six sides?

Write out the first 12 or so powers of 4 : $$ 4,16,64,256,1024,4096,16384,65536, \ldots $$ Now create two sequences: the sequence of final digits: \(4,6,4,6,4,6, \ldots\) the sequence of leading digits: \(4,1,6,2,1,4,1,6, \ldots\) Both sequences seem to consist of a single "block", which repeats over and over for ever. (a) How long is the apparent repeating block for the first sequence? How long is the apparent repeating block for the second sequence? (b) It may not be immediately clear whether either of these sequences really repeats forever. Nor may it be clear whether the two sequences are alike, or whether one is quite different from the other. Can you give a simple proof that one of these sequences recurs, that is, repeats forever? (c) Can you explain why the other sequence seems to recur, and decide whether it really does recur forever?

(a) The operation of "squaring" is a function: it takes a single real number \(x\) as input, and delivers a definite real number \(x^{2}\) as output. \- Every positive number arises as an output ("is the square of something" ). \(-\) Since \(x^{2}=(-x)^{2},\) each output (other than 0 ) arises from at least two different inputs. \- If \(a^{2}=b^{2},\) then \(0=a^{2}-b^{2}=(a-b)(a+b)\), so either \(a=b\), or \(a=-b\). Hence no two positive inputs have the same square, so each output (other than 0 ) arises from exactly two inputs (one positive and one negative). \- Hence each positive output \(y\) corresponds to just one positive input, called \(\sqrt{y}\). Find: (i) \(\sqrt{49}\) (ii) \(\sqrt{144}\) (iii) \(\sqrt{441}\) (iv) \(\sqrt{169}\) (v) \(\sqrt{196}\) (vi) \(\sqrt{961}\) (vii) \(\sqrt{96100}\) (b) Let \(a>0\) and \(b>0\). Then \(\sqrt{a b}>0\), and \(\sqrt{a} \times \sqrt{b}>0\), so both expressions are positive. Moreover, they have the same square, since $$ (\sqrt{a b})^{2}=a b=(\sqrt{a})^{2} \cdot(\sqrt{b})^{2}=(\sqrt{a} \times \sqrt{b})^{2} $$ \(\therefore \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}\) Use this fact to simplify the following: (i) \(\sqrt{8}\) (ii) \(\sqrt{12}\) (iii) \(\sqrt{50}\) (iv) \(\sqrt{147}\) (v) \(\sqrt{288}\) (vi) \(\sqrt{882}\) (c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example - namely the regular pentagon. Suppose that a regular pentagon \(A B C D E\) has sides of length \(1 .\) (i) Prove that the diagonal \(A C\) is parallel to the side \(E D\). (ii) If \(A C\) and \(B D\) meet at \(X,\) explain why \(A X D E\) is a rhombus. (iii) Prove that triangles \(A D X\) and \(C B X\) are similar. (iv) If \(A C\) has length \(x\), set up an equation and find the exact value of \(x\). (v) Find the exact length of \(B X\). (vi) Prove that triangles \(A B D\) and \(B X A\) are similar. (vii) Find the exact values of \(\cos 36^{\circ}, \cos 72^{\circ}\). (viii) Find the exact values of \(\sin 36^{\circ}, \sin 72^{\circ}\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.