Interactive Audio Lesson
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Applying Pythagoras' Theorem
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Today we're going to apply Pythagoras' Theorem to real-world situations! Remember, it states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Can anyone tell me what that looks like geometrically?
It's like a triangle where we can use 'a squared plus b squared equals c squared' to find the longest side!
Exactly right! Let's look at our ladder problem: the ladder reaches 12 meters high and is 5 meters away at the base. How would we set this up using Pythagoras' Theorem?
We can denote the ladder as 'c', the height as 'a', and the base as 'b'. So we would solve 12² + 5² = c².
Exactly! Can you calculate the length of the ladder?
Sure! That gives us 144 + 25 = 169, so c = √169 which is 13 meters!
Great job! Remember that checking our calculations helps reinforce understanding. Let's summarize: the Pythagorean theorem is essential when calculating lengths in right triangles.
Finding Missing Sides in Triangles
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Moving forward, let’s find a missing side length in triangle ABC where ∠B is 90°, and we know AB = 7 cm and AC = 25 cm. How do we start?
We can use the converse of Pythagoras' Theorem to check if it's a right triangle, but since we already know it is, we can just apply the theorem directly.
Correct! Can anyone calculate BC for us?
Yes! So BC² = AC² − AB² = 625 − 49, which gives us 576, meaning BC = √576, which is 24 cm!
Nice work! Remembering that we’re subtracting the square of the shorter side helps avoid confusion. We can apply the Triangle Sum Theorem to validate our results by checking if the angles add up to 180°.
Understanding Angle of Elevation with Trigonometry
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Let’s switch gears and talk about the angle of elevation. We have a 15 m tall pole casting a shadow of 20 m. How can we find the angle of elevation?
We can use the tangent function! Tangent is opposite over adjacent, right?
Absolutely! So what would our calculation look like?
It's tan θ = 15/20, which simplifies to 0.75. To find the angle, we can use the arctan function.
Exactly! What does using tan⁻¹(0.75) give us?
That means θ ≈ 36.87°. We can also visualize this with a right triangle where the height of the pole is our opposite side.
That's perfect! Always remember to connect how we visualize these triangles! Great work, everyone!
Introduction & Overview
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Quick Overview
Standard
In this section, we explore worked examples that demonstrate the application of geometric and trigonometric theorems, including calculating lengths and angles, thereby solidifying students' understanding through practical application.
Detailed
Detailed Summary
In the 'Worked Examples' section, we delve into practical applications of geometry and trigonometry through structured problems designed to highlight the key theorems discussed earlier in the chapter. Key examples include:
- Ladder Problem: Applying Pythagoras' Theorem to find the length of a ladder leaning against a wall, where the wall's height and base distance from the wall are given. This example emphasizes the relationship between height, base, and hypotenuse in right triangles.
- Triangle ABC: Another example involving triangle ABC where given the lengths of two sides, we apply Pythagoras' Theorem to find a missing side length. This example reinforces deductive reasoning and calculation skills crucial for solving geometric problems.
- Angle of Elevation: This example explores the use of trigonometric ratios to calculate the angle of elevation from a shadow's length cast by a pole. By understanding the roles of sine, cosine, and tangent, students connect theoretical knowledge to practical scenarios.
These examples serve as essential learning tools for students, allowing them to practice key mathematical concepts in real-life contexts. Students are encouraged to reason logically and apply these concepts in exercises following the examples.
Audio Book
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Example 1: Ladder Against a Wall
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Example 1:
A ladder is leaning against a wall. It reaches a height of 12 m on the wall and the base is 5 m away from the wall. Find the length of the ladder.
Solution:
Using Pythagoras' Theorem:
Ladder² = 12² + 5² = 144 + 25 = 169
Ladder = √169 = 13 m
Detailed Explanation
In this example, we are trying to find the length of a ladder that is leaning against a wall. Here, the height the ladder reaches on the wall is considered the 'opposite' side of a right triangle, and the distance from the base of the ladder to the wall is the 'adjacent' side. By applying Pythagoras' Theorem, which states that in a right triangle, the square of the hypotenuse (the ladder in this case) is equal to the sum of the squares of the other two sides, we calculate the length of the ladder. We square the height (12 m) and the base (5 m), add those two results, and then take the square root to find the hypotenuse, which we find to be 13 m.
Examples & Analogies
Imagine you're trying to put up a Christmas tree that needs to be secured against a wall. If you measure how high the tree reaches the wall and the distance it is from the wall, you can use this same principle to determine how tall the tree stand needs to be to keep it upright.
Example 2: Finding a Missing Side
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Example 2:
In triangle ABC, ∠B = 90°, AB = 7 cm, AC = 25 cm. Find BC.
Solution:
Using Pythagoras' Theorem:
BC² = AC² − AB² = 625 − 49 = 576
BC = √576 = 24 cm
Detailed Explanation
In this triangle, we are given a right angle at B and the lengths of the sides AB and AC. We need to find the length of side BC. Using Pythagoras' Theorem, we can work backwards since we know the hypotenuse (AC), and one side (AB). By squaring the hypotenuse and the known side and then subtracting the square of the known side from the square of the hypotenuse, you can solve for the missing side. Finally, taking the square root of the result gives us the length of side BC, which turns out to be 24 cm.
Examples & Analogies
Think of a soccer field where you know the distance from the goal to a certain point in the field (the hypotenuse) and from there to the sideline (one leg of the triangle). You want to find out how far to the center of the goal you are — this is represented by the second leg of the triangle, and you can find it using this theorem.
Example 3: Angle of Elevation
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Example 3:
Find the angle of elevation if a 15 m tall pole casts a shadow of 20 m.
Solution:
Use tan θ = Opposite / Adjacent = 15 / 20 = 0.75
θ = tan⁻¹(0.75) ≈ 36.87°
Detailed Explanation
In this example, we are trying to determine the angle of elevation from the end of a shadow to the top of a pole. The height of the pole (15 m) is the 'opposite' side, and the length of the shadow (20 m) represents the 'adjacent' side. The tangent function relates these two sides through the angle θ we want to find. By setting up the equation tan(θ) = opposite/adjacent, we can solve for the angle by taking the arctan (tan⁻¹) of the ratio 15/20. This calculation tells us the angle of elevation, which we find to be approximately 36.87 degrees.
Examples & Analogies
Picture standing next to a tall building and looking up at its roof. The angle you look up forms an 'angle of elevation' from where you're standing to the top of the building. The height of the building and the distance you are standing from it creates a triangle, similar to how we determine angles using triangles in math.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Pythagorean Theorem: A fundamental relationship in right triangles used to find missing side lengths.
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Angle of Elevation: The angle measured from the horizontal up to an object.
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Trigonometric Ratios: Ratios of the angles to the lengths of sides in right triangles.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Finding the length of a ladder using Pythagoras' Theorem when the ladder reaches 12 m high and is 5 m away from the wall.
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Finding the length of side BC in triangle ABC with AB = 7 cm and AC = 25 cm when ∠B = 90°.
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Finding the angle of elevation when a 15 m pole casts a 20 m shadow.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In a triangle, two sides squared, give you the hypotenuse that’s compared.
📖 Fascinating Stories
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Imagine a right triangle named Righty, always hiking up a ladder to reach the sky, measuring angles and lengths, climbing high, solving problems one by one, as he reaches for sun.
🧠 Other Memory Gems
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SOH CAH TOA: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), Tangent (Opposite/Adjacent).
🎯 Super Acronyms
PATH
- Pythagorean theorem
- Angle of elevation
- Trigonometric ratios
- Heights.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Pythagoras' Theorem
Definition:
A fundamental relation in Euclidean geometry between the three sides of a right triangle, stating that the square of the hypotenuse equals the sum of the squares of the other two sides.
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Term: Angle of Elevation
Definition:
The angle formed by the horizontal line and the line of sight from an observer to an object above the horizontal.
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Term: Trigonometric Ratios
Definition:
Ratios that relate the angles of a right triangle to the ratios of its sides—sine, cosine, and tangent.
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Term: Hypotenuse
Definition:
The longest side of a right triangle, opposite the right angle.