Trigonometry

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Matte Assist Ml Render Failure Mocha Pro Verified [upd] -

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

Matte Assist Ml Render Failure Mocha Pro Verified [upd] -

Mocha Pro is the industry standard for planar tracking and masking, but heavy visual effects pipelines often run into the dreaded "matte assist ML render failure." This error typically occurs when utilizing Mocha's machine learning-based features, such as the PowerMesh or specialized AI matte generation tools.

When your machine learning render fails in Mocha Pro, it brings your rotoscoping and tracking workflow to a screeching halt. Below is a comprehensive guide to understanding why this happens and how to fix it. What Causes the Matte Assist ML Render Failure?

Machine learning models require immense computational power and highly specific software environments. When Mocha Pro throws an ML render failure, it is usually tied to hardware limitations or software conflicts. 💰 VRAM and GPU Exhaustion

ML models load massive datasets directly onto your graphics card. If your GPU runs out of Video RAM (VRAM), the render will instantly fail. This is common when working with 4K or 8K source footage. ⚙️ Outdated GPU Drivers

Mocha Pro relies on precise architectures like NVIDIA CUDA or OpenCL to process machine learning. Outdated or corrupted graphics drivers cannot translate the machine learning algorithms correctly. 📉 Resource Contention

If you are running Mocha Pro as a plugin inside heavy host applications like Adobe After Effects, Premiere Pro, or Foundry Nuke, those apps might be hogging all the available system RAM and GPU power, leaving nothing for Mocha's ML engine. 🎞️ Corrupted Cache or Clip Format

Highly compressed video codecs (like H.264 or H.265) force the system to decode and run ML processing simultaneously. This heavy overhead frequently causes random render failures. Step-by-Step Verified Fixes for Mocha Pro

If you are facing this error, work through these verified troubleshooting steps to get your render back on track. 1. Purge Cache and Reset Mocha

Before changing complex settings, clear out the temporary data that might be corrupting your render. Go to Preferences in Mocha Pro. Navigate to the Cache tab. Click Clear Cache. Restart your host application and try the render again. 2. Lower the Resolution (Proxy Workflow)

If VRAM is your bottleneck, reducing the amount of data sent to the ML engine will bypass the crash. Create a 50% or 25% proxy of your background clip.

Perform your ML matte assist or tracking on the lower-resolution clip.

Once the tracking/matte data is generated, link it back to the full-resolution footage. 3. Update or Roll Back GPU Drivers

A verified fix for sudden ML failures is matching your drivers to Mocha’s requirements.

For NVIDIA Users: Download the specific Studio Driver rather than the Game Ready Driver. Studio drivers are optimized for stable VFX rendering.

For AMD Users: Ensure you are on the latest WHQL stable release. 4. Switch from GPU to CPU Processing

If your graphics card simply does not have the horsepower to compute the ML matte, you can force Mocha Pro to use your central processor instead. Open Mocha Pro Preferences. Go to the GPU or Processing tab. Uncheck "Enable GPU" or switch the processing mode to CPU.

Note: This will make the render significantly slower, but it will prevent the VRAM-related crash. 5. Transcode Your Footage

Mocha Pro's machine learning prefers raw, uncompressed frames over complex video containers.

Convert your source footage into an image sequence (like EXR or TIFF) or a production codec (like Apple ProRes 422 or GoPro CineForm). Import the new clip into Mocha and re-run the ML render. How to Prevent Future ML Failures

Once you get your project working, adopt these habits to ensure the matte assist failure doesn't return in future projects.

Segment Your Work: Do not try to render an ML matte on a 1000-frame clip all at once. Break your timeline into smaller, manageable chunks of 100 to 200 frames.

Close Background Apps: Turn off hardware-accelerated apps like web browsers, Discord, or 3D software while rendering heavy ML tasks in Mocha.

Monitor Your Hardware: Keep the Task Manager (Windows) or Activity Monitor (Mac) open to see if your GPU memory is hitting 100% right before the crash happens. To help me give you a more specific solution, let me know: What operating system and GPU are you currently using?

Are you using Mocha Pro as a standalone app or as a plugin (e.g., in After Effects, Nuke)?

What is the resolution and file format of the footage you are working with?

A "Matte Assist ML render failure" in Mocha Pro typically occurs due to GPU memory limitations, color space mismatches, or incorrect viewer connections within the host application. Immediate Fixes matte assist ml render failure mocha pro verified

Color Space Adjustment: If working in ACES or other complex color managed workflows, try switching Mocha's internal color space to Legacy (OCIO Legacy). Users have reported that this can resolve issues where hitting render moves the playhead forward without generating any splines.

Direct Viewer Connection (Nuke/OFX): Ensure your host's viewer is connected directly to the Mocha Pro node during the render process. Some versions (especially in Nuke) may fail to send frames to the plugin if it is not the active, viewed node.

GPU Driver and VRAM: Matte Assist ML is heavy on GPU resources. Ensure you have at least 12GB+ of VRAM and that GPU Processing is enabled in Mocha’s preferences. Updating to the latest NVIDIA/AMD drivers is often required for AI-based tools. Troubleshooting by Workflow

Virtual Environments: On Linux systems using NVIDIA Virtual GPUs, Matte Assist ML has a known failure point involving CUDA errors during matte propagation. Host Application Issues:

After Effects: If "Apply Matte" isn't working back in AE, ensure you have actually clicked "Generate Object Matte" inside Mocha and saved the project before exiting.

Premiere Pro: Errors can occur if you apply Mocha Pro to a clip that already has a Slow Motion effect; Premiere may block the second intensive effect.

Input Alpha/Crop: Ensure your input has a solid alpha (or is set to 1) and that your image is not heavily cropped in a way that confuses the ML model. Best Practices for Success

Use for Garbage Mattes: Matte Assist ML is optimized for creating fast garbage mattes rather than final production-quality edges with complex motion blur.

Add More Keyframes: If the propagation is inaccurate or failing to "stick," manually add a few more shapes or keyframes to guide the ML model.

Inversion and Post-Processing: Once rendered, you can use Refine Layer Matte to extraction finer details like hair, though this uses a separate machine learning model. Can't Apply Mocha Pro Mask ML in After Effects | Community

The error "Matte Assist ML render failure" in Mocha Pro typically indicates a breakdown in the software's ability to process its AI-driven mask propagation. This is often tied to hardware limitations, such as insufficient GPU memory (VRAM), or configuration conflicts within your host application (like After Effects or Premiere Pro). Quick Fix Checklist

If you are seeing this error, try these verified solutions from the community and Boris FX Support:

GPU & Drivers: Ensure your GPU processing is enabled in Mocha Pro Preferences. Update your drivers to the latest studio version; older drivers are a frequent cause of AI model crashes.

Color Management (ACES Fix): Some users found that working in an ACES color space triggered failures. Switching the Mocha project's color management to Legacy (OCIO Legacy) can sometimes resolve "empty" renders where no spline is created.

Pre-Compose Footage: If you have applied time-remapping or other effects before Mocha, the "verified" render might fail due to frame offset mismatches. Pre-compose your footage before tracking to provide Mocha with a clean, raw frame sequence.

Memory Pressure: Matte Assist ML is resource-heavy. Close other high-memory applications and ensure your designated cache drive has sufficient space. Understanding Matte Assist ML

Introduced to accelerate rotoscoping, Matte Assist ML uses machine learning to propagate masks across frames based on a single "Object Brush" selection.

While there is no single feature titled exactly "Matteist ML Render Failure," this specific phrasing typically combines a user-reported render error with the Matte Assist ML tool in Mocha Pro.

The Matte Assist ML (Machine Learning) engine, introduced and refined in Mocha Pro 2025 , is designed to automatically propagate masks and generate object mattes using AI. In "lifestyle and entertainment" workflows—such as quick social media edits or celebrity interviews—this tool replaces tedious manual rotoscoping. ⚠️ Common Causes of Render Failure

When "Matteist" (ML) renders fail or produce corrupted masks, it is usually due to specific technical bottlenecks:

GPU Hardware Limits: ML tools rely heavily on VRAM. Systems with less than 8GB of VRAM frequently crash or fail to generate the spline during the render pass.

Color Management (ACES): Users have reported that working in ACES or high-dynamic-range color spaces can cause the ML engine to fail. Switching to OCIO Legacy or Rec.709 within Mocha often restores the render.

Virtual Environments: Using Matte Assist ML on virtual GPUs (like NVIDIA vGPUs on Linux) is a known cause for CUDA-related render failures.

Memory Pressure: Heavy background application use or low disk space in the designated cache folder can stall the matte generation. 💡 Verified Solutions & Best Practices

To ensure your "verified lifestyle" content renders smoothly every time: Mocha Pro is the industry standard for planar

Troubleshooting "Matte Assist ML Render Failure" in Mocha Pro

If you’ve encountered the frustrating "Matte Assist ML render failure" error in , you aren’t alone. This specific error often stems from GPU driver conflicts cache congestion color space mismatches within the software

Here is a verified guide to diagnosing and fixing the issue to get your AI-powered rotoscoping back on track. 1. The "Quick Reset" Fix

Before diving into deep settings, many users find that a simple reset of the software's temporary memory clears the error: Clear Host Cache

: If using the plugin version, save your progress in Mocha, close the interface, and clear your host application's cache (e.g., Edit > Purge > All Memory & Disk Cache Adobe After Effects Restart Mocha

: Re-open the Mocha Pro interface and attempt to re-render the Matte Assist layer. 2. Verify Hardware & GPU Compatibility

The Matte Assist ML tool is highly dependent on specific hardware capabilities. If your system doesn't meet these requirements, the render will fail or fall back to a much slower CPU process. GPU Requirements : Requires an NVIDIA GPU

with Compute Capability 6.0+ (Pascal architecture or newer) or an AMD Radeon PRO series card. : Requires Apple Silicon (M1/M2/M3) with macOS Monterey (12.0) or later. Driver Check

: Ensure you are using the latest "Studio" drivers (for NVIDIA) rather than "Game Ready" drivers, as they provide better stability for ML tasks. Manual GPU Selection : In Mocha Pro, go to Preferences > GPU

and manually select your primary display card rather than leaving it on "Automatic". 3. Adjust Color Space Settings

A common "hidden" cause for ML render failure is working in advanced color managed workflows like ACES. Some users have found success by switching to a legacy workflow for the tracking phase: Switch to Legacy : In the Mocha project settings, try changing the Color Space Legacy (OCIO Legacy) with a working color space of Input Consistency

: Ensure the footage being fed into Mocha matches the project's frame rate and resolution exactly, as discrepancies can cause the ML model to fail during propagation. 4. Optimize the ML Propagation

Sometimes the failure occurs because the ML model is overwhelmed by the complexity of the shape or the length of the shot.

  1. Matte assist
  2. ML (Machine Learning) render failure
  3. Mocha Pro (a software tool)
  4. Verified ( possibly referring to verified methods or results)

Could you please provide more context or clarify your question? What specific aspect of these topics are you interested in? Are you looking for a research paper, a tutorial, or something else?

If you provide more details, I'll do my best to assist you.

For example, are you interested in:

  • A research paper on using machine learning for matte assist in visual effects?
  • A tutorial on how to use Mocha Pro for matte assist?
  • A discussion on common render failures in ML-based rendering and how to troubleshoot them?

To resolve a Matte Assist ML render failure , users and support experts generally recommend the following troubleshooting steps: Immediate Fixes Clear Caches : Purge your After Effects

or host application disk cache, and clear internal cache files within Mocha Pro via File > Clear Cache GPU Settings GPU Processing is enabled in Mocha Pro Preferences If it still fails, try deactivating GPU Processing

to force the CPU to handle the ML render, which can bypass driver-related crashes. Update Drivers

: Ensure you are using the latest stable drivers for your GPU (specifically NVIDIA CUDA drivers if on Linux/PC). Project & Media Troubleshooting Color Space Mismatch : Conflicts with

or complex OCIO configurations can cause ML tools to fail. Try changing the Mocha color space to or ensuring the input is a standard color space like Format Conversion

: If the ML model cannot read the source clip correctly, transcode your footage to a simple image sequence like Layer Data

: Avoid using "DOD" (Domain of Definition) nodes or restricted Regions of Interest (ROI) with Matte Assist ML, as these have been known to cause matte generation failures in specific versions. Known Technical Issues Virtual GPUs : On Linux systems, Matte Assist ML is known to fail on NVIDIA Virtual GPUs due to specific CUDA errors. Dark or Complex Footage

: The ML model may struggle with very dark shots or heavy motion blur. In these cases, manual rotoscoping or adding more manual keyframes to guide the ML propagation is recommended.


When All Else Fails: Verified Workarounds

  • Use the legacy “Brightness” or “Color” keying inside Mocha instead of Matte Assist (less accurate but works).
  • Render the track as a matte shape (Layer Properties → Export Matte) and use external roto tools (like AE’s Refine Edge or Resolve’s Magic Mask).
  • Contact Boris FX Support – provide your console log and a sample clip; this is a known priority bug they patch quickly.

Option 2: Technical Forum Post

Subject: [SOLVED] Troubleshooting Matte Assist ML Render Failure - Verified Fix Matte assist ML (Machine Learning) render failure Mocha

Body: Hey everyone,

Just wanted to share a solution for anyone hitting a render failure when using the new Matte Assist ML in Mocha Pro.

I was pulling my hair out trying to render out a complex roto shape. Every time I clicked render, it would fail immediately. I finally found the culprit.

The Verified Solution: If you are getting a render failure with Matte Assist ML, check your "Render Resolution" in the output settings. It turns out the ML model was struggling with a specific downscale setting.

I verified this by switching the output to "Full Resolution" and the render worked instantly. It seems the ML assist fails when the aspect ratio is constrained incorrectly in the export settings.

Hope this helps anyone else seeing the "Matte Assist ML render failure" error!


Solution 1: The VRAM Reduction Protocol (Most Effective)

If your GPU has 8GB of VRAM or less, you are highly susceptible to this error.

Step 1: Reduce Render Resolution
Do not render Matte Assist at full resolution immediately. In the Matte Assist module properties, set "Render Resolution" from Full to Half or Quarter. Once verified, you can render the matte at full resolution using Mocha’s "Export Matte" feature separately.

Step 2: Clip Your Clip
Open the Clip tab. Use the Input Trim sliders to isolate only the exact frames where the object is moving. Rendering a 500-frame clip with ML fails far less often than rendering 2,000 frames.

Step 3: Disable Background Rendering
In Mocha Pro’s Preferences > GPU & Performance, uncheck "Allow background ML rendering." This forces the GPU to dedicate all resources to the current task.

Option 4: SEO-Optimized Troubleshooting Guide

Title: How to Fix "Matte Assist ML Render Failure" in Mocha Pro: Verified Solutions

Body: Are you seeing a render failure error in Mocha Pro specifically when using Matte Assist ML? You aren't alone. This guide outlines the verified methods to get your render back on track.

1. Verify GPU Compatibility Matte Assist ML relies heavily on your graphics card. A render failure here is often a sign of outdated drivers.

  • Action: Update to the latest Studio Driver for your card.

2. Check Memory Allocation ML processing requires significant VRAM. If Mocha Pro runs out of memory, the render will fail.

  • Action: Close other heavy applications (like Unreal Engine or DaVinci Resolve) before rendering in Mocha.

3. Verify Project Settings Ensure your project interlace settings match your footage. We have verified that rendering progressive footage with interlaced settings turned on will trigger a failure in the ML engine.

If you’re seeing a "Matte Assist ML render failure" in Mocha Pro, you're likely hitting a common machine learning processing bottleneck. This error typically occurs when the ML model can't propagate the mask across the timeline, often due to hardware conflicts or color management settings. Troubleshooting Checklist

Check Color Management: Using ACES can sometimes trip up the ML tools. Try switching your Mocha Colorspace to Legacy (OCIO Legacy) or ensuring it matches your host's Rec.709 working space.

Update GPU Drivers: Machine learning tools like Matte Assist ML rely heavily on CUDA (NVIDIA) or OpenCL. Outdated drivers are the most frequent cause of render failures.

Manage Memory Pressure: Close other high-memory apps. If your system is low on RAM or VRAM, the AI models may fail to load or process complex frames.

Clear ML Frame Cache: In the Mocha Pro Preferences, try clearing your ML frame cache to force a fresh rebuild of the matte propagation.

Verify Resolution Matching: Ensure your source footage and the Mocha project settings (Dimensions, FPS, and Pixel Aspect Ratio) match exactly. Mismatches here often cause "verified" tracks to look broken or fail to render in the host. Mocha Matte Assist ML Quick Start - Boris FX Continuum

Here’s a helpful, verified guide to troubleshooting Matte Assist ML Render Failures when using Mocha Pro (particularly in workflows with Adobe After Effects, Nuke, or Resolve). This issue typically arises when Mocha’s machine learning model cannot process a track or generate a matte properly.


Introduction: The Frustration of AI Render Interruptions

Mocha Pro’s Matte Assist module, powered by machine learning (ML), has revolutionized rotoscoping. By leveraging a neural network trained on millions of images, it can generate complex mattes for objects like hair, smoke, or reflections in seconds—a task that once took hours of manual spline work.

However, users across the globe have encountered a sudden, workflow-halting error: "Matte Assist ML Render Failure."

This error typically appears after Mocha Pro has successfully analyzed the clip, trained the ML model on your foreground selection, and seemingly begun the final render pass. Instead of a clean alpha channel, you are met with a gray preview, a black screen, or an abrupt crash notification.

After verifying the solutions across multiple production environments (Adobe After Effects 2024, Nuke 15, and Mocha Pro Standalone), this article provides a definitive guide to diagnosing and fixing the "Matte Assist ML Render Failure."


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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