Unit 7: Right Triangles and Trigonometry – Homework 1 Answer Key
Right triangles are the cornerstone of trigonometry. So they let us relate angles to side lengths, a skill that is essential for geometry, physics, engineering, and everyday problem‑solving. This article presents a comprehensive answer key for Unit 7, Homework 1, covering all the problems that students typically encounter. Which means alongside each solution, we explain the underlying concepts, provide step‑by‑step reasoning, and highlight common pitfalls. By the end, you should understand why each answer is correct and how to apply the same logic to similar problems Simple, but easy to overlook..
Introduction
The first homework set in a unit on right triangles usually tests the following core ideas:
- Basic definitions – hypotenuse, legs, acute angles.
- Pythagorean theorem – relationship between the three sides.
- Trigonometric ratios – sine, cosine, tangent for acute angles.
- Inverse trigonometric functions – determining angles from ratios.
- Special right triangles – 30°–60°–90° and 45°–45°–90°.
This answer key walks through each problem, showing the calculations, the reasoning, and the final results. Keep in mind that the values are rounded to two decimal places unless otherwise specified Simple as that..
Problem 1 – Identify the Hypotenuse
Question
In a right triangle, the legs measure 6 cm and 8 cm. Which side is the hypotenuse?
Solution
The hypotenuse is always the side opposite the right angle and is the longest side.
Using the Pythagorean theorem:
[ c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} ]
So the hypotenuse is 10 cm.
Common mistake: Forgetting that the hypotenuse is the longest side; some students think the smallest side is the hypotenuse.
Problem 2 – Apply the Pythagorean Theorem
Question
Find the length of the missing side in a right triangle where the hypotenuse is 13 cm and one leg is 5 cm.
Solution
Let the unknown leg be (b).
[
13^2 = 5^2 + b^2 \
169 = 25 + b^2 \
b^2 = 144 \
b = 12 \text{ cm}
]
Answer: 12 cm.
Tip: Always square the known sides before adding or subtracting Surprisingly effective..
Problem 3 – Solve for an Angle Using Sine
Question
A right triangle has legs of 9 cm (adjacent to angle ( \theta )) and 12 cm (opposite to ( \theta )). Find ( \theta ) And it works..
Solution
Use the tangent ratio because we have opposite and adjacent sides:
[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{9} = \frac{4}{3} ]
[ \theta = \arctan!\left(\frac{4}{3}\right) \approx 53.13^\circ ]
Answer: ( \theta \approx 53.13^\circ ).
Common error: Mixing up opposite and adjacent; always check which side is adjacent to the angle in question.
Problem 4 – Find a Missing Side Using Cosine
Question
In a right triangle, the hypotenuse is 10 m and one acute angle is (30^\circ). What is the length of the side adjacent to that angle?
Solution
Cosine relates the adjacent side to the hypotenuse:
[ \cos 30^\circ = \frac{\text{adjacent}}{10} ]
[ \text{adjacent} = 10 \times \cos 30^\circ \approx 10 \times 0.8660 = 8.66 \text{ m} ]
Answer: ( \approx 8.Here's the thing — 66 \text{ m} ). Tip: Remember that ( \cos 30^\circ = \sqrt{3}/2 ) Simple, but easy to overlook..
Problem 5 – Special 45°–45°–90° Triangle
Question
A 45°–45°–90° triangle has a hypotenuse of 14√2 cm. What is the length of each leg?
Solution
In a 45°–45°–90° triangle, legs are equal, and the hypotenuse equals leg × √2.
Let leg = (x):
[ x\sqrt{2} = 14\sqrt{2} \ x = 14 \text{ cm} ]
Answer: Each leg is 14 cm.
Common pitfall: Forgetting that the legs are equal in this special triangle.
Problem 6 – Special 30°–60°–90° Triangle
Question
A 30°–60°–90° triangle has a short leg of 5 cm. Find the length of the long leg and the hypotenuse.
Solution
Properties of a 30°–60°–90° triangle:
- Short leg (opposite 30°) = (x)
- Long leg (opposite 60°) = (x\sqrt{3})
- Hypotenuse = (2x)
Given short leg (x = 5) cm:
[ \text{Long leg} = 5\sqrt{3} \approx 8.66 \text{ cm} ] [ \text{Hypotenuse} = 2 \times 5 = 10 \text{ cm} ]
Answers: Long leg ≈ 8.So 66 cm; Hypotenuse = 10 cm. Note: Many students mistakenly swap the long and short legs Easy to understand, harder to ignore..
Problem 7 – Apply the Law of Sines in a Right Triangle
Question
In a right triangle, angle ( \alpha = 45^\circ ) and side opposite ( \alpha ) is 7 cm. Find the length of the hypotenuse And it works..
Solution
Because the triangle is right, we can use the sine ratio:
[ \sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{h} ]
[ h = \frac{7}{\sin 45^\circ} = \frac{7}{\sqrt{2}/2} = 7 \times \frac{2}{\sqrt{2}} = 7\sqrt{2} \approx 9.90 \text{ cm} ]
Answer: ( h \approx 9.90 \text{ cm} ).
Tip: Recognize that for a 45° angle in a right triangle, the opposite side equals the adjacent side That's the part that actually makes a difference..
Problem 8 – Verify a Right Triangle Using Pythagorean Theorem
Question
Check whether a triangle with sides 9 cm, 12 cm, and 15 cm is right-angled.
Solution
Sort sides so that the largest is the hypotenuse candidate (15 cm):
[ 9^2 + 12^2 = 81 + 144 = 225 \ 15^2 = 225 ]
Since (9^2 + 12^2 = 15^2), the triangle is a right triangle It's one of those things that adds up..
Answer: Yes, it is a right triangle.
Common mistake: Treating the smallest side as the hypotenuse.
Problem 9 – Find an Angle Using the Tangent Function
Question
A ladder leans against a wall, forming a 75° angle with the ground. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?
Solution
Use the tangent ratio:
[ \tan 75^\circ = \frac{\text{opposite}}{3} \ \text{opposite} = 3 \times \tan 75^\circ \approx 3 \times 3.732 = 11.20 \text{ m} ]
Answer: Approximately 11.Day to day, 20 m high. Tip: Remember that a large angle close to 90° will produce a large opposite side Which is the point..
Problem 10 – Determine a Missing Side Using the Sine Function
Question
A right triangle has a 30° angle and the side opposite this angle is 4 m. Find the length of the hypotenuse.
Solution
[
\sin 30^\circ = \frac{4}{h} \
h = \frac{4}{0.5} = 8 \text{ m}
]
Answer: 8 m.
Common error: Confusing the sine ratio with the cosine ratio.
FAQ – Common Questions About Right Triangles
| Question | Answer |
|---|---|
| What is the longest side in a right triangle? | The hypotenuse, opposite the right angle. |
| Can a right triangle have an obtuse angle? | No. So the other two angles must be acute (less than 90°). On the flip side, |
| **How do I remember the trigonometric ratios? ** | Mnemonic SOHCAHTOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. |
| **When do I use inverse trig functions?Still, ** | When you know a ratio and need to find the corresponding angle. In real terms, |
| **What are the side ratios in a 30°–60°–90° triangle? ** | Short leg : Long leg : Hypotenuse = 1 : √3 : 2. |
| What are the side ratios in a 45°–45°–90° triangle? | Leg : Leg : Hypotenuse = 1 : 1 : √2. |
Conclusion
Mastering right triangles and trigonometry requires a blend of memorization (special triangle ratios), procedural skill (applying the Pythagorean theorem and trigonometric ratios), and conceptual understanding (knowing why each step works). By systematically working through the problems above, students build confidence in solving a wide range of related questions. So remember to keep the definitions fresh, practice the ratios, and always double‑check which side corresponds to which angle. With these tools, you’ll be well‑prepared for the next unit and any real‑world applications that demand trigonometric insight.
