Special Right Triangles - 30 60 90 - Geometry & Trigonometry | SAT Math

The Organic Chemistry Tutor

7 Nov 201711:11

EducationalLearning

32 Likes 10 Comments

TLDRThis educational video script introduces the properties of a 30-60-90 right triangle, explaining the relationship between its angles and side lengths. It emphasizes that the side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle is the 30-degree side multiplied by the square root of three. The script provides step-by-step examples to calculate missing sides, highlighting the need to multiply or divide based on whether one is moving from a smaller to a larger angle or vice versa. It also includes a sample SAT math problem involving a circle and a 30-60-90 triangle, demonstrating how to find the area of a shaded region by subtracting the triangle's area from the circle's area, resulting in an exact answer and a decimal approximation.

Takeaways

📐 The 30-60-90 triangle is a special right triangle where the sides are in the ratio 1:√3:2.
🔍 If you know the side opposite the 30-degree angle, you can find the hypotenuse by multiplying by 2.
🔢 Conversely, if you know the hypotenuse, you can find the side opposite the 30-degree angle by dividing by 2.
📈 To find the side opposite the 60-degree angle, multiply the side opposite the 30-degree angle by √3.
📉 If you have the side opposite the 60-degree angle and need to find the side opposite the 30-degree angle, divide by √3.
🌐 The longest side of a 30-60-90 triangle is opposite the largest angle (90 degrees), and the shortest side is opposite the smallest angle (30 degrees).
🔄 To find a longer side, multiply the known side by the appropriate factor (2 or √3), and to find a shorter side, divide.
📚 The script provides several examples demonstrating how to calculate missing sides of a 30-60-90 triangle using the given ratios.
📉 The script explains that to find the height of a 30-60-90 triangle, you take half of the hypotenuse.
📏 To find the base of the triangle, multiply the height by √3.
🎯 The script concludes with an example of calculating the area of a shaded region in a circle with a 30-60-90 triangle, emphasizing the use of the triangle's properties to find the necessary lengths.

Q & A

What is the significance of the 30-60-90 right triangle in geometry?
-The 30-60-90 right triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. It has a consistent ratio of side lengths, which are 1 (for the side opposite the 30-degree angle), √3 (for the side opposite the 60-degree angle), and 2 (for the hypotenuse), making it useful for solving various geometric problems.
How do you find the side length across the 90-degree angle if you know the side across the 30-degree angle in a 30-60-90 triangle?
-If you know the side length across the 30-degree angle, you can find the side length across the 90-degree angle (the hypotenuse) by multiplying the known side length by 2.
What operation should you perform to find the side length across the 30-degree angle if you know the hypotenuse of a 30-60-90 triangle?
-To find the side length across the 30-degree angle when you know the hypotenuse, you should divide the hypotenuse by 2.
How can you determine the side length across the 60-degree angle if you have the side across the 30-degree angle?
-To find the side length across the 60-degree angle from the side across the 30-degree angle, multiply the known side length by √3.
What is the relationship between the sides of a 30-60-90 triangle when going from a smaller angle to a larger angle?
-When going from a smaller angle to a larger angle in a 30-60-90 triangle, you need to multiply the side length by the appropriate factor: multiply by 2 to go from the 30-degree angle to the 90-degree angle, and multiply by √3 to go from the 30-degree angle to the 60-degree angle.
How do you find the missing sides of a 30-60-90 triangle if you are given the side across the 60-degree angle?
-If you are given the side across the 60-degree angle, you can find the side across the 30-degree angle by dividing by √3, and then find the hypotenuse by multiplying the side across the 30-degree angle by 2.
What is the longest side of a 30-60-90 triangle and why is it the longest?
-The longest side of a 30-60-90 triangle is the hypotenuse, which is opposite the 90-degree angle. It is the longest because in any right triangle, the hypotenuse is always the longest side, and in a 30-60-90 triangle, the hypotenuse is twice the length of the side opposite the 30-degree angle.
Can you find the area of a 30-60-90 triangle without knowing the lengths of all three sides?
-No, to find the area of a 30-60-90 triangle, you need to know the lengths of at least two sides. The area can be calculated as one-half the product of the base and the height, which in a 30-60-90 triangle are the sides opposite the 30-degree and 60-degree angles, respectively.
How does the video script use the properties of a 30-60-90 triangle to solve for the area of a shaded region in a circle?
-The script uses the properties of a 30-60-90 triangle to find the lengths of the sides of the triangle, which are then used to calculate the area of the triangle. This area is subtracted from the area of the circle (π times the radius squared) to find the area of the shaded region.
What is the formula for the area of a circle and how is it used in the context of the video script?
-The formula for the area of a circle is πr², where r is the radius of the circle. In the context of the video script, the diameter of the circle is given as 20 units, so the radius is half of that, 10 units. The area of the circle is then calculated as π times 10 squared, which is 100π square units.

Outlines

00:00

📐 Introduction to 30-60-90 Right Triangle

In this video, we will discuss the 30-60-90 right triangle. It is important to first watch the video on the 45-45-90 triangle. For the 30-60-90 triangle, the side opposite the 30-degree angle is 1, the hypotenuse is 2, and the side opposite the 60-degree angle is 1 times the square root of 3. Key relationships: multiplying by 2 to find the hypotenuse, dividing by 2 to find the side opposite the 30-degree angle, multiplying by the square root of 3 to find the side opposite the 60-degree angle, and dividing by the square root of 3 when going backwards. The longest side is opposite the largest angle and the shortest side is opposite the smallest angle.

05:00

🧮 Example Problem 1

Example problem: Given a 30-60-90 triangle with the side opposite the 30-degree angle as 5 units, find the other sides. The hypotenuse is found by multiplying by 2 (5 * 2 = 10). The side opposite the 60-degree angle is found by multiplying by the square root of 3 (5 * √3 = 5√3).

10:02

📏 Example Problem 2

Example problem: Given a 30-60-90 triangle with the side opposite the 30-degree angle as 12√3, find the other sides. The hypotenuse is found by multiplying by 2 (12√3 * 2 = 24√3). The side opposite the 60-degree angle is found by multiplying by the square root of 3 (12 * 3 = 36).

🔢 Example Problem 3

Example problem: Given a 30-60-90 triangle with the hypotenuse as 12, find the other sides. The side opposite the 30-degree angle is found by dividing by 2 (12 / 2 = 6). The side opposite the 60-degree angle is found by multiplying by the square root of 3 (6 * √3 = 6√3).

📐 Example Problem 4

Example problem: Given a 30-60-90 triangle with the hypotenuse as 16√3 and the side opposite the 60-degree angle as 30, find the other sides. The side opposite the 30-degree angle is found by dividing by the square root of 3 (16√3 / √3 = 16). The side opposite the 60-degree angle is found by multiplying by the square root of 3 (16 * 3 = 48).

🔎 Example Problem 5

Example problem: Given a 30-60-90 triangle with the side opposite the 60-degree angle as 14√3, find the other sides. The side opposite the 30-degree angle is found by dividing by the square root of 3 (14√3 / √3 = 14). The hypotenuse is found by multiplying by 2 (14 * 2 = 28).

📝 SAT Math Example

Example problem: Given a 30-60-90 triangle with the hypotenuse as 20, find the area of the shaded region. The side opposite the 30-degree angle is found by dividing by 2 (20 / 2 = 10). The side opposite the 60-degree angle is found by multiplying by the square root of 3 (10 * √3 = 10√3). The area of the triangle is 1/2 * base * height = 1/2 * 10√3 * 10 = 50√3. The radius of the circle is 10, so the area of the circle is π * 10^2 = 100π. The area of the shaded region is 100π - 50√3, which is approximately 227.6 square units.

Mindmap

Keywords

💡30-60-90 Triangle

A 30-60-90 triangle is a special type of right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees. The sides are in a fixed ratio: the side opposite the 30-degree angle is the shortest and is considered the '1' part, the side opposite the 60-degree angle is '√3' times the shortest side, and the hypotenuse (opposite the 90-degree angle) is twice the shortest side. In the video, this concept is fundamental as it provides the basis for solving various problems involving the triangle's sides and angles.

💡Hypotenuse

The hypotenuse is the longest side of a right triangle, which is opposite the right angle (90 degrees). In the context of the video, the hypotenuse is twice the length of the side opposite the 30-degree angle. It is a key element in the 30-60-90 triangle as it helps in determining the lengths of the other two sides using the specific ratios.

💡Side Ratios

In a 30-60-90 triangle, the side ratios are crucial for determining the lengths of the sides. The script explains that the side opposite the 30-degree angle is 1 part, the hypotenuse is 2 parts, and the side opposite the 60-degree angle is √3 times the shortest side. These ratios are used throughout the video to solve for unknown sides.

💡Multiplication and Division

The video script emphasizes the use of multiplication and division to find the lengths of the sides of a 30-60-90 triangle. If you know the side opposite the 30-degree angle and want to find the hypotenuse, you multiply by 2. Conversely, to find the side opposite the 30-degree angle from the hypotenuse, you divide by 2. To find the side opposite the 60-degree angle from the 30-degree side, you multiply by √3, and to find the 30-degree side from the 60-degree side, you divide by √3.

💡Square Root of Three

The square root of three (√3) is a mathematical constant that is used in the context of the 30-60-90 triangle to find the length of the side opposite the 60-degree angle. In the script, it is mentioned that this value is used when multiplying or dividing the side opposite the 30-degree angle to find the corresponding side opposite the 60-degree angle.

💡Longest Side

The longest side of a 30-60-90 triangle is the hypotenuse, which is opposite the largest angle, the 90-degree angle. The script explains that when moving from a smaller angle to a larger angle, you need to multiply the side lengths, indicating that the hypotenuse is the longest side.

💡Shortest Side

The shortest side in a 30-60-90 triangle is opposite the smallest angle, which is the 30-degree angle. The script mentions that when moving from a larger angle to a smaller angle, you need to divide the side lengths, which implies that the shortest side is opposite the 30-degree angle.

💡Area Calculation

The area calculation is a key concept in the video, especially when dealing with the shaded region in a circle with a 30-60-90 triangle. The area of a circle is calculated using πr², where 'r' is the radius, and the area of a right triangle is calculated as 1/2 * base * height. The script uses these formulas to find the area of the shaded region by subtracting the area of the triangle from the area of the circle.

💡Diameter

The diameter of a circle is a straight line that passes through the center and whose endpoints lie on the circle. In the script, it is mentioned that the segment AC, which is the hypotenuse of the 30-60-90 triangle, is also the diameter of the circle. This is important because the diameter is twice the radius, which helps in finding the radius and subsequently the area of the circle.

💡SAT Math Question

The SAT Math section is a standardized test that assesses a student's mathematical abilities. The script includes an example of a typical SAT Math question involving a 30-60-90 triangle and a circle, which tests the application of geometric principles and formulas to find the area of a shaded region. This example illustrates the practical application of the concepts discussed in the video.

Highlights

Introduction to the 30-60-90 right triangle and its side ratios.

Comparison with the 45-45-90 triangle video for a comprehensive understanding.

Explanation of side values across the 30, 60, and 90 degree angles in a 30-60-90 triangle.

Rule of multiplying by two to find the hypotenuse from the 30-degree side.

Rule of dividing by two to find the 30-degree side from the hypotenuse.

Method to find the 60-degree side by multiplying the 30-degree side by the square root of three.

Inverse method of dividing the 60-degree side by the square root of three to find the 30-degree side.

Observation that the longest side is opposite the largest angle, and vice versa.

Multiplication rule for transitioning from a smaller angle side to a larger angle side.

Division rule for transitioning from a larger angle side to a smaller angle side.

Example problem solving with a given 30-degree side and calculating the other two sides.

Second example with a given side across the 60-degree angle, finding the hypotenuse and 30-degree side.

Third example with the hypotenuse given, calculating the 30-degree and 60-degree sides.

Fourth example with the side across the 60-degree angle given, finding the other two sides.

Fifth example involving rationalizing the denominator when dividing by the square root of three.

Final example with the side across the 60-degree angle given, solving for the 30-degree and hypotenuse sides.

Application of 30-60-90 triangle properties in a SAT math question involving a circle and a triangle.

Calculation of the area of a shaded region by subtracting the area of a triangle from a circle's area.

Conversion of the exact answer into a decimal to provide a numerical solution for the area.

Transcripts

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