Trigonometry

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Onlyfans Alice Redlips Aka Emanovak Pack E [upd] May 2026

THERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles right triangle. They are special because with simple geometry we can know the ratios of their sides, and therefore solve any such triangle.

Theorem.  In a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3.

30-60-90 triangle

We will prove that below.

Note that the smallest side, 1, is opposite the smallest angle, 30°; while the largest side, 2, is opposite the largest angle, 90°.  (Theorem 6). (For, 2 is larger than square root of 3.  Also, while 1 : square root of 3: 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1 : 2 : square root of 3 easier to remember.)

The cited theorems are from the Appendix, Some theorems of plane geometry.

Here are examples of how we take advantage of knowing those ratios.  First, we can evaluate the functions of 60° and 30°.

Example 1.   Evaluate cos 60°.

Answer.  For any problem involving a 30°-60°-90° triangle, the student should not use a table.  The student should sketch the triangle and place the ratio numbers.

30-60-90 triangle

Since the cosine is the ratio of the adjacent side to the hypotenuse, we can see that

cos 60° = ½.

Example 2.   Evaluate sin 30°.

Answer.   According to the property of cofunctions, sin 30° is equal to cos 60°.   sin 30° = ½.

On the other hand, you can see that directly in the figure above.

Problem 1.   Evaluate sin 60° and tan 60°.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

The sine is the ratio of the opposite side to the hypotenuse.

    sin 60° =  square root of 3
 2
 = ½square root of 3.

Lesson 5 of Algebra

The tangent is ratio of the opposite side to the adjacent.

    tan 60° =  square root of 3
 1
= square root of 3.

Problem 2.   Evaluate cot 30° and cos 30°.

1 : 2: SqRt 3

The cotangent is the ratio of the adjacent side to the opposite.

   Therefore, on inspecting the figure above, cot 30° =  square root of 3
 1

= square root of 3.

Or, more simply, cot 30° = tan 60°.

Problem 1

As for the cosine, it is the ratio of the adjacent side to the hypotenuse. Therefore,

cos 30° =  square root of 3
 2
 = ½square root of 3.

Before we come to the next Example, here is how we relate the sides and angles of a triangle:

Triangle ABC

If an angle is labeled capital A, then the side opposite will be labeled small a.  Similarly for angle B and side b, angle C and side c.

Example 3.   Solve the right triangle ABC if angle A is 60°, and side AB is 10 cm.

1:2:SqRt3

Solution.  To solve a triangle means to know all three sides and all three angles.  Since this is a right triangle and angle A is 60°, then the remaining angle B is its complement, 30°.

Again, in every 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : square root of 3, as shown on the left.

When we know the ratios of the sides, then to solve a triangle we do not require the trigonometric functions or the Pythagorean theorem. We can solve it by the method of similar figures.

Now, the sides that make the equal angles are in the same ratio. Proportionally,

2 : 1 = 10 : AC.

2 is two times 1. Therefore 10 is two times AC.  AC is 5 cm.

The side adjacent to 60°, we see, is always half the hypotenuse.

As for BC—proportionally,

2 : square root of 3 = 10 : BC.

To produce 10,  2 has been multiplied by 5. Therefore, square root of 3 will also be multiplied by 5.  BC is 5square root of 3 cm.

In other words, since one side of the standard triangle has been multiplied by 5, then every side will be multiplied by 5.

1:2:SqRt3

1 : 2 : square root of 3 = 5 : 10 : 5square root of 3.

Compare Example 11 here.

Again: When we know the ratio numbers, then to solve the triangle the student should use this method of similar figures, not the trigonometric functions.

(In Topic 10, we will solve right triangles whose ratios of sides we do not know.)

Problem 3.   In the right triangle DFE, angle D is 30° and side DF is 3 inches.  How long are sides d and f ?

1 : 2: Sq rt 3

1 : 2: SqRt 3

   The student should draw a similar triangle in the same orientation. Then see that the side corresponding to square root of 3 was multiplied by square root of 3.

Lesson 26 of Algebra

Therefore, each side will be multiplied by square root of 3. Side d will be
1timessquare root of 3 = square root of 3. Side f will be 2square root of 3.

Problem 4.   In the right triangle PQR, angle P is 30°, and side r is 1 cm.  How long are sides p and q ?

1 : 2: SqRt 3

1 : 2: SqRt 3

The side corresponding to 2 has been divided by 2.   Therefore, each side must be divided by 2.  Side p will be ½, and side q will be ½square root of 3.

Problem 5.   Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm.

30-60-90 triangle

30-60-90 triangle

The side adjacent to 60° is always half of the hypotenuse -- therefore, side b is 9.3 cm.
But this is the side that corresponds to 1. And it has been multiplied by 9.3.  Therefore, side a will be multiplied by 9.3.
It will be 9.3square root of 3 cm.

Problem 6.    Prove:  The area A of an equilateral triangle whose side is s, is

A = ¼square root of 3s2.

30-60-90 triangle

The area A of any triangle is equal to one-half the sine of any angle times the product of the two sides that make the angle. (Topic 2, Problem 6.)

In an equilateral triangle each side is s , and each angle is 60°.  Therefore,

A = ½ sin 60°s2.

Since sin 60° = ½square root of 3,

Problem 1

A = ½· ½square root of 3 s2 = ¼square root of 3s2.

Problem 7.    Prove:  The area A of an equilateral triangle inscribed in a circle of radius r, is

A 3
4
square root of 3 r2.

30-60-90 triangle

Onlyfans Alice Redlips Aka Emanovak Pack E [upd] May 2026

The request for an "informative paper" on a specific OnlyFans content "pack" refers to a collection of digital media distributed via the OnlyFans subscription platform. Identity and Platform Alice Redlips (aka Emanovak)

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The Power of Personal Brand: Lessons from Alice Redlips In the modern digital landscape, your online presence is your "digital resume". Whether you are a seasoned professional or just starting, the way you curate your content can significantly impact your career trajectory. Taking inspiration from the aesthetic and influence of figures like Alice Redlips

, we explore how to blend style with substance to build a career-defining social media presence. 1. Define Your Signature Aesthetic

Just as a specific "red lip kit" or a signature look can define a creator's brand, your social media needs a cohesive visual identity. Consistency is Key

: Choose a style that reflects your professional "vibe" while remaining authentic to your personality. Visual Assets

: Use high-quality profile pictures—like a clear headshot or an action shot—to immediately communicate your discipline and focus to potential collaborators or employers. 2. Treat Your Socials as a Living Portfolio Your channels shouldn't just show you do; they should show you do it. Showcase Impact

: Instead of saying you "created content," highlight that your content "increased engagement by 42%" or "reached 1.2 million impressions". Document the Process

: Share behind-the-scenes glimpses of your projects, lessons learned, or "do’s and don’ts" in your industry. This positions you as an expert rather than just a participant. 3. Strategy Over Luck

Success in content creation isn't accidental; it’s the result of strategic planning. Batch Your Content

: To avoid burnout and maintain a consistent posting schedule, create and schedule your posts in batches. The 80/20 Rule

: Dedicate 80% of your content to adding value—sharing industry insights or articles—and 20% to self-promotion or "humble bragging" about your achievements.

Alice Redlips is a lifestyle and beauty content creator known for her digital presence on platforms like Instagram, TikTok, and Alice Red Lips - OFTV. Her career focuses on personal branding, style inspiration, and professional reporting within the entertainment industry. Content and Professional Focus

Beauty and Style: Alice positions her platforms as spaces for inspiration and tips designed to help her audience feel "confident and beautiful inside and out". This includes makeup tutorials featuring products from brands like Dior and Charlotte Tilbury.

Red Carpet Reporting: Beyond standard influencer content, she has a background as an artist and reporter, specifically conducting Red Carpet Interviews in Hollywood.

Digital Reach: She maintains an active presence with over 150 reels on Instagram and significant engagement on TikTok, where her handle @alice.redlips is frequently tagged in beauty-related community content. Career Themes

Advocacy for Non-Traditional Paths: Alice emphasizes that there is "never one path" in the media industry. She supports the transition of social media creators into traditional acting or reporting roles, noting that consistent online content often serves as the "foot in the door".

Entrepreneurial Journey: She often shares motivational content about the "long nights, sacrifices, and hard work" required to sustain a career in Hollywood and digital media.

Title: Unveiling the Enigmatic Alice Redlips: A Deep Dive into her OnlyFans Journey

Introduction

In the vast and ever-evolving world of adult content creation, few personalities have managed to carve out a niche as distinct and captivating as Alice Redlips, also known as Emanovak. With her undeniable charm, tantalizing content, and a loyal following, Alice has emerged as a significant figure within the OnlyFans community. This blog post aims to peel back the layers of her online persona, exploring her journey, the allure of her content, and what makes her 'pack e' so eagerly anticipated by her fans.

The Rise of Alice Redlips on OnlyFans

OnlyFans, a platform known for its adult content creators, has been the launching pad for many individuals seeking to monetize their allure and charisma. Alice Redlips, whose real name is Emanovak, has leveraged this platform to build a substantial following. Her rise to prominence can be attributed to her unique blend of tantalizing visuals, engaging personality, and a keen understanding of what her audience desires.

Content and Appeal

Alice Redlips' content is a carefully curated mix of photos and videos that showcase her in various states of undress and intimacy. Her 'pack e' – a term that seems to refer to specific bundles or collections of her content – has become particularly popular among her fans. These packs often feature high-quality images and videos, meticulously produced to offer a premium experience to her subscribers.

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The Allure of 'Pack E'

The anticipation for 'pack e' among Alice Redlips' fans can be attributed to several factors. Firstly, the exclusivity of the content. These packs often feature material that is not available elsewhere, making them highly sought after. Secondly, the production quality. Each pack is crafted with attention to detail, ensuring that subscribers receive content that meets their high expectations.

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Conclusion

Alice Redlips, or Emanovak, has undoubtedly made a mark on OnlyFans with her captivating presence and enticing content. Her journey on the platform is a testament to her appeal and the evolving landscape of adult content creation. As she continues to grow her following and expand her offerings, including the highly anticipated 'pack e', it's clear that Alice Redlips is a name to watch in the world of adult entertainment.

Whether you're a long-time fan or just discovering her, Alice Redlips' OnlyFans journey offers insights into the power of personal branding, audience engagement, and the monetization of allure in the digital age.

Disclaimer: This blog post is intended for informational purposes and does not endorse or promote any specific adult content. It aims to provide an overview of Alice Redlips' career on OnlyFans based on publicly available information.

The search results do not provide a specific career biography or content creator profile for an individual named "Alice Redlips."

There is an Instagram search category titled "What Happened To Alice Redlips," but the associated content largely features posts about established fashion figures like Anna Wintour and general industry trends rather than a dedicated profile for a creator by that name.

The phrase "Alice Redlips" may be a niche handle, a pseudonym, or a combination of terms. In the context of social media careers:

Red Lips as Symbolism: Red lipstick is frequently used as a brand identity or symbol of power and rebellion in fashion and beauty content.

Brand Roles: Companies like Alice + Olivia frequently hire Social Content Creators to manage their digital presence on TikTok and Instagram, typically offering salaries between $70,000 and $90,000 for New York-based roles.

General Career Path: Most digital content producers build their careers by specializing in high-quality short-form video (Reels/TikToks), maintaining a cohesive visual identity, and consistently engaging with trends to grow a brand's online presence.

Early Beginnings and Rise to Fame

Alice Redlips, whose real name is not publicly known, started her social media journey on platforms like Instagram and TikTok. She initially gained attention for her captivating lip-sync videos, dance challenges, and lifestyle content. Her charisma and creativity quickly resonated with audiences, leading to a significant increase in followers.

Content Style and Niche

Alice Redlips is recognized for her:

  • Sassy and Humorous Content: She often creates comedic skits, parodies, and reactions that showcase her personality and entertaining side.
  • Beauty and Fashion Focus: Her content frequently features makeup tutorials, product reviews, and fashion hauls, which have helped her build a strong presence in the beauty and fashion niches.
  • Lifestyle and Vlogging: Alice also shares aspects of her personal life, including travel vlogs, daily routines, and behind-the-scenes glimpses into her content creation process.

Career Milestones and Collaborations

As her social media presence grew, Alice Redlips began to collaborate with:

  • Brands and Influencer Marketing Agencies: She has partnered with various beauty and fashion brands, promoting their products and services to her engaged audience.
  • Other Social Media Personalities: Alice has collaborated with fellow influencers, content creators, and celebrities on sponsored content, challenges, and projects.
  • Content Creation and Production: She has expanded her team to produce high-quality content, including music videos, short films, and web series.

Business Ventures and Entrepreneurial Spirit

Alice Redlips has leveraged her online fame to launch:

  • Merchandise Lines: She sells branded merchandise, such as clothing, accessories, and beauty products, to her loyal fan base.
  • Beauty and Lifestyle Products: Alice has introduced her own line of beauty products, including skincare items, makeup, and hair care tools.

Impact and Influence

Alice Redlips' social media presence and career have made a notable impact on:

  • Young Audiences: Her content has inspired and entertained millions of young people, who look up to her as a role model and source of creativity.
  • The Beauty and Fashion Industries: Alice's influence has contributed to the growth and diversification of the beauty and fashion industries, promoting inclusivity, body positivity, and self-expression.

Challenges and Controversies

Like many social media personalities, Alice Redlips has faced:

  • Criticism and Backlash: She has encountered criticism for certain content, collaborations, or statements, which she has addressed through apologies and explanations.
  • Mental Health and Burnout: Alice has spoken publicly about the pressures of maintaining a social media presence, managing her mental health, and finding work-life balance.

Overall, Alice Redlips is a talented and influential social media personality who has built a successful career through her creative content, entrepreneurial spirit, and dedication to her audience.

Alice Redlips is a digital content creator and social media personality primarily known for her presence on platforms like OFTV, Instagram, and TikTok. Her career and content focus on beauty, lifestyle, and fashion, often characterized by a "passionate lover of all things beautiful and inspiring". Social Media Content & Themes

Beauty and Tutorials: Alice frequently shares makeup inspiration and "Get Ready With Me" (GRWM) routines. Her content often features specific beauty products, such as matte lipsticks and hydration-focused concealers, and explores unique aesthetics like Korean-inspired makeup vibes.

Style and Inspiration: Her OFTV channel aims to provide tips and knowledge to help followers feel confident and beautiful, often centered around "style, beauty, and adventure".

Travel and Vlogging: She produces travel-related content, including a series on a trip to the Adirondacks, which mixes lifestyle vlogging with visual storytelling. onlyfans alice redlips aka emanovak pack e

Aesthetic Presence: Her Instagram and TikTok presence is marked by a focus on visual trends, such as high-contrast red-lip looks that match her handle. Career Overview

Alice’s career is rooted in the "influencer-to-creator" model, leveraging multiple social media platforms to build a personal brand.

Platform Diversification: She maintains an active presence across major social apps, including Instagram and Facebook.

Brand Collaborations: While specific long-term corporate roles aren't public, she is frequently cited in the context of modern social media trends where creators are cast in various projects based on their follower reach and unique visual identity.

Professional Identity: She identifies as a content creator who seeks to "create a world of style" for her audience.

Here’s a draft social media post / career update for Alice Redlips, written in a professional yet engaging tone. You can adapt it for LinkedIn, Instagram, Twitter, or a personal website.


Option 1: LinkedIn / Professional Bio Update

🎬 Career Update: Alice Redlips

I’m excited to share that I’m leveling up my focus at the intersection of social media content strategy and sustainable career growth.

Over the past [X years], I’ve helped brands and creators build authentic digital presences — but I realized one thing was missing: a clear roadmap for you, the person behind the posts.

Now, I’m doubling down on:

📌 Social Media Content – Strategy, storytelling, and systems that actually work (without burning out).
📌 Career Clarity – Helping creators & digital professionals turn content into long-term opportunities, partnerships, and income.

Whether you’re launching your first channel or pivoting your creative career, I’m here to bridge the gap between “going viral” and building something that lasts.

Let’s connect if you’re ready to stop guessing and start growing — on purpose.

🔗 [Link to portfolio / booking / newsletter]

#AliceRedlips #ContentStrategy #CreatorEconomy #CareerDevelopment #SocialMediaManagement


Option 2: Instagram / TikTok caption (casual, confident, direct)

New chapter loading 🚀

I’m Alice Redlips — and from now on, this space is all about social media content + career strategy.

No fluff. No burnout culture. Just real systems to help you: ✅ Create content with purpose
✅ Attract opportunities (not just likes)
✅ Build a career you don’t need a vacation from

If you’ve ever felt stuck between being a “content machine” and actually growing your career — stick around.

Drop a 🔴 if you’re ready to take both seriously.

#AliceRedlips #ContentThatConverts #CareerGrowth #CreatorTips


Option 3: Short bio for social media profiles (Twitter / IG / TikTok bio)

Alice Redlips ✨
Social media content + career strategy
Helping creators turn posts into paychecks & purpose.
📩 Collab / consult: [link]


Alice Redlips (often stylized as Alice Red Lips) is a lifestyle and beauty content creator known for her presence on visual and video-first platforms. Her career serves as a case study for building a personal brand centered around aesthetics, travel, and wellness. Core Content Pillars

Alice's content strategy revolves around high-quality visual storytelling focused on three primary areas:

Travel & Adventure: She documents extensive road trips and nature excursions, including multi-part series on the Adirondacks, Washington, and New York City.

Beauty & Style: Her brand emphasizes "all things beautiful," offering inspiration and tips aimed at helping her audience feel confident and stylish.

Wellness & Mindfulness: She integrates self-care content such as gentle home yoga, meditation sessions, and animal-focused experiences like visiting alpaca farms. Platform & Career Strategy

Alice utilizes a multi-platform approach to maximize her reach and diversify her content delivery:

OFTV (OnlyFans TV): She uses this platform to host longer-form, curated video series that go beyond standard social media clips, such as detailed travelogues.

Instagram: Like many modern creators, she uses Instagram for visual snapshots and community engagement through hashtags like #aliceredlips.

Persona Management: Her career reflects a "curated reality" approach, balancing personal interests with a public persona designed to inspire her followers. Lessons from Her Creator Path The request for an "informative paper" on a

For those looking to follow a similar career trajectory, Alice's growth highlights key industry practices:

Iterative Improvement: Early content often acts as a learning phase; successful creators like Alice focus on "showing up and improving over time" rather than waiting for perfection.

Niche Authority: By focusing on a specific aesthetic—often centered on the "red lip" signature—she creates a recognizable brand identity.

Audience Engagement: She positions her channel as a collaborative space where followers "create a world of style" together, fostering a sense of community.

The digital creator economy has transformed how individuals build brands and connect with audiences. When analyzing the growth of online personalities who use multiple monikers or personas, several key factors in digital branding and audience engagement come into play. The Dynamics of Digital Personas

In the modern landscape of social media and subscription-based platforms, creators often utilize different names or "aka" labels to segment their content or transition between different niches. This strategy allows for:

Brand Versatility: Maintaining separate identities can help a creator appeal to different demographics simultaneously.

Search Engine Optimization: Using multiple recognizable names increases the likelihood of appearing in diverse search queries.

Content Evolution: Personas allow creators to experiment with new styles—from lifestyle blogging to high-fashion photography—without alienating their original core audience. The Concept of Content Curation

The mention of specific "packs" or collections in search trends highlights a shift in how digital media is consumed. Modern audiences often seek curated experiences. From a branding perspective, organizing work into "packs" or series helps in:

Thematic Consistency: Grouping content by aesthetic or production style.

Archival Value: Providing a chronological look at a creator's professional development.

Monetization Strategies: Offering bundled content as a way to provide value to long-term supporters. Navigating the Creator Economy

The rise of platforms that allow for direct-to-consumer interaction has given creators unprecedented agency over their work. For those studying this field, the focus is often on how creators maintain "stickiness" or high retention rates. Success in this area is typically driven by:

Direct Engagement: Moving beyond passive posting to active community management.

Aesthetic Branding: Developing a signature look—such as a specific color palette or recurring visual motif—that makes the content immediately recognizable.

Platform Diversification: Using various social channels to funnel traffic to a primary hub. Conclusion

The interest in specific digital creators and their curated collections reflects the high demand for personalized, high-quality digital identities. Understanding the mechanics of how these creators organize their work and manage their reputations provides valuable insight into the future of online media and independent entrepreneurship.

Supporting creators through official and verified channels remains the most effective way to ensure the sustainability of the creative industry and the safety of the digital ecosystem.

As a content creator, Alice Redlips leverages her platforms to share insights, tips, and personal experiences related to:

  1. Social Media Content Strategy: She likely shares advice on how to create compelling content, grow a following, and engage with audiences effectively.
  2. Career Development: Her content may include guidance on professional growth, job search strategies, and skill development.
  3. Lifestyle and Wellness: She might also share aspects of her personal life, focusing on wellness, self-care, and maintaining a balanced lifestyle.

If you're interested in learning more about Alice Redlips or looking for advice on creating social media content and advancing your career, here are some general tips:

1. Find Your Visual Anchor

For Alice, it is the lipstick. For you, it might be a hat, a specific piece of jewelry, or a lighting setup. This anchor becomes your brand's unique selling proposition (USP).

Character Profile: Alice Redlips

Name: Alice Redlips (Stage Name) Real Name: Alice Vance Occupation: Jazz Vocalist / Private Investigator Setting: A neo-noir cityscape, perpetual rain, neon lights.

Description: Alice is known for her signature bright red lipstick and a voice that sounds like "honey poured over gravel." By night, she sings at the Velvet Lounge, but her performances are often a cover for her true work: gathering secrets for the city's most desperate clients. She uses her charm and presence to slip into places the police can't reach.

Vignette: The Drop

The rain tapped a frantic rhythm against the window of the Velvet Lounge, drowning out the hiss of the espresso machine in the corner. Alice sat in the back booth, her crimson coat a splash of blood against the dark leather. She checked her watch.

"You’re late," she said, not looking up as a shadow fell across the table.

"Traffic was murder," the man replied, sliding into the seat opposite her. He wore a trench coat that looked like it had seen better decades. He placed a small, unassuming pack on the table. "This is everything you asked for. Photos, letters, the whole history."

Alice finally looked up, her eyes sharp. She picked up the pack, weighing it in her hand. It felt heavier than paper should. "And the payment?"

"Half now, half on delivery," the man rasped.

Alice reached into her clutch, her movements slow and deliberate. She placed a stack of bills on the table. "Get out of here before the set starts. If anyone sees us..."

"I know the drill, Redlips."

As he vanished back into the night, Alice opened the pack. Inside wasn't just correspondence; it was a ledger that could bring down half the city council. She smiled, her red lips curving upward. It was going to be a long night, but at least she had the truth.

4. #LipService (Q&A, community hot takes)

Alice answers fan questions about burnout, negotiation, hate comments, or platform growth.
🗣 Format: “Sip or spill” – holds a coffee cup, red lip stain on rim, gives unfiltered opinions. Sassy and Humorous Content : She often creates


3. Build a Community, Not a Fanbase

Alice Redlips interacts with her top-tier fans through private channels. She treats them as patrons of the arts, not just numbers on a screen. This loyalty translates to recession-proof income.

4. Charge for the "How," not the "What"

Anyone can show a lipstick. Alice charges for how to apply it for a corporate headshot, how to negotiate a raise while wearing it, and how to build a content calendar that accommodates it.

Problem 8.   Prove:  The angle bisectors of an equilateral triangle meet at a point that is two thirds of the distance from the vertex of the triangle to the base.

30-60-90 triangle

Let ABC be an equilateral triangle, let AD, BF, CE be the angle bisectors of angles A, B, C respectively; then those angle bisectors meet at the point P such that AP is two thirds of AD.

First, triangles BPD, APE are congruent.

For, since the triangle is equilateral and BF, AD are the angle bisectors, then angles PBD, PAE are equal and each 30°;
and the side BD is equal to the side AE, because in an equilateral triangle the angle bisector is the perpendicular bisector of the base.

Theorem 2

Angles PDB, AEP then are right angles and equal.

Therefore,

Angle-side-angle

triangles BPD, APE are congruent.

30-60-90 triangle

  Now,  BP
PD
= csc 30° = 2.

Problem 2

Therefore, BP = 2PD.

But AP = BP, because triangles APE, BPD are conguent, and those are the sides opposite the equal angles.
Therefore, AP = 2PD.
     Therefore AP is two thirds of the whole AD.
Which is what we wanted to prove.

The proof

Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3.  It is based on the fact that a 30°-60°-90° triangle is half of an equilateral triangle.

Draw the equilateral triangle ABC.  Then each of its equal angles is 60°.  (Theorems 3 and 9)

A 30-60-90 triangle

Draw the straight line AD bisecting the angle at A into two 30° angles.
Then AD is the perpendicular bisector of BC  (Theorem 2).  Triangle ABD therefore is a 30°-60°-90° triangle.

Now, since BD is equal to DC, then BD is half of BC.

This implies that BD is also half of AB, because AB is equal to BC.  That is,

BD : AB = 1 : 2

From the Pythagorean theorem, we can find the third side AD:

AD2 + 12 = 22
AD2 = 4 − 1 = 3
AD = square root of 3.

Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : square root of 3; which is what we set out to prove.

Corollary.  The square drawn on the height of an equalateral triangle is three fourths of the square drawn on the side.


Next Topic:  The Isosceles Right Triangle


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