Right Triangle Test Right Triangle Mastery
Unit 5: Geometric Relationships & Trigonometry
Name:
Date:
Instructions: Complete all sections. For word problems, show all steps, including the initial setup, calculations, and final units. Round all decimal answers to the nearest hundredth unless otherwise specified.
Section 1: Pythagorean Theorem & Converse
1. A construction crew is checking the squareness of a foundation. They measure two sides as 9 feet and 12 feet. What must the diagonal measure be for the corner to be a perfect right angle?
15 feet
18 feet
21 feet
25 feet
2. Determine if a triangle with side lengths of 8, 15, and 17 is a right triangle. Justify your answer using the Converse of the Pythagorean Theorem.
3. Find the missing side length \(x\) in the right triangle shown. Provide the exact answer in simplest radical form.
\(x = \)
12 x 13
Section 2: Special Right Triangles
4. In a \(45^\circ-45^\circ-90^\circ\) triangle, the hypotenuse measures \(14\sqrt{2}\) cm. Find the length of each leg.
Leg Length: ____________________
\(14\sqrt{2}\) \(45^\circ\)
5. For the \(30^\circ-60^\circ-90^\circ\) triangle shown, find the values of \(y\) and \(z\). Give exact answers.
\(y = \) ________________ \(z = \) ________________
y z 10 \(30^\circ\)
Section 3: Trigonometric Ratios
6. Based on the triangle below, which ratio represents \(\cos B\)?
A B C 20 21 29
\(\frac{20}{29}\)
\(\frac{21}{29}\)
\(\frac{20}{21}\)
\(\frac{21}{20}\)
7. Solve for the measure of angle \(x\) to the nearest degree.
x 15 8
Workspace:
\(x \approx\) ____________________
Section 4: Real-World Applications
8. A 24-foot ladder is leaning against a vertical wall. The base of the ladder is 7 feet away from the base of the wall. What is the measure of the angle the ladder makes with the ground?
Show Your Work
9. An observer at the top of a 150-meter lighthouse spots a ship in the distance. The angle of depression to the ship is \(18^\circ\). How far is the ship from the base of the lighthouse?
Show Your Work
Right Triangle Test Answer Key Right Triangle Mastery
Teacher Answer Key
Total Points: 100
Grading Rubric
Multiple Choice: 5 pts each. Short Answer: 10 pts each. Applications: 15 pts each. Partial credit should be awarded for correct setup even if calculation errors occur.
Section 1: Pythagorean Theorem & Converse
1. Answer: 15 feet
Work: \(9^2 + 12^2 = c^2 \Rightarrow 81 + 144 = 225 \Rightarrow c = \sqrt{225} = 15\)
2. Answer: Yes, it is a right triangle.
Justification: \(8^2 + 15^2 = 64 + 225 = 289\). Since \(17^2 = 289\), the equation \(a^2 + b^2 = c^2\) is satisfied.
3. Answer: \(x = 5\)
Work: \(x^2 + 12^2 = 13^2 \Rightarrow x^2 + 144 = 169 \Rightarrow x^2 = 25 \Rightarrow x = 5\)
Section 2: Special Right Triangles
4. Answer: 14 cm
Work: In a \(45-45-90\) triangle, \(hypotenuse = leg \cdot \sqrt{2}\). So, \(14\sqrt{2} = leg \cdot \sqrt{2}\), which means \(leg = 14\).
5. Answer: \(y = 5\); \(z = 5\sqrt{3}\)
Work: In a \(30-60-90\) triangle, the short leg \(y\) is half the hypotenuse (\(10/2 = 5\)). The long leg \(z\) is the short leg times \(\sqrt{3}\) (\(5\sqrt{3}\)).
Section 3: Trigonometric Ratios
6. Answer: \(\frac{21}{29}\)
Reasoning: \(\cos B = \frac{adjacent}{hypotenuse} = \frac{BC}{AB} = \frac{21}{29}\).
7. Answer: \(x \approx 28^\circ\)
Work: \(\tan x = \frac{8}{15} \Rightarrow x = \tan^{-1}(\frac{8}{15}) \approx 28.07^\circ\).
Section 4: Real-World Applications
8. Answer: \(\approx 73.04^\circ\)
Step 1 (Setup): Identify \(hypotenuse = 24\) and \(adjacent = 7\).
Step 2 (Equation): \(\cos \theta = \frac{7}{24}\)
Step 3 (Solve): \(\theta = \cos^{-1}(\frac{7}{24}) \approx 73.039...\)
9. Answer: \(\approx 461.65\) meters
Step 1 (Setup): Angle of depression = Angle of elevation = \(18^\circ\). Height (\(opposite\)) = 150m. Distance (\(adjacent\)) = \(d\).
Step 2 (Equation): \(\tan(18^\circ) = \frac{150}{d}\)
Step 3 (Solve): \(d = \frac{150}{\tan(18^\circ)} \approx \frac{150}{0.3249} \approx 461.652...\)
Right Triangle Review Slides Right Triangle
Mastery
Comprehensive Exam Review
The Pythagorean Theorem
\[a^2 + b^2 = c^2\]
Only applies to Right Triangles
c is ALWAYS the hypotenuseHypotenuse is the longest side
a (leg) b (leg) c (hypotenuse)
The Converse
If the sum of the squares of the two shorter sides equals the square of the longest side, then the triangle is a Right Triangle.
Example 1: (5, 12, 13)
\(5^2 + 12^2 = 25 + 144 = 169\)
\(13^2 = 169 \rightarrow \text{RIGHT}\)
Example 2: (4, 5, 8)
\(4^2 + 5^2 = 16 + 25 = 41\)
\(8^2 = 64 \rightarrow \text{NOT RIGHT}\)
Special: 45-45-90
Ratio Rule
\(x : x : x\sqrt{2}\)
Legs are congruent (Isosceles)
Hypotenuse = \(Leg \times \sqrt{2}\)
x x \(x\sqrt{2}\)
Special: 30-60-90
x \(x\sqrt{3}\) 2x \(30^\circ\)
Ratio Rule
\(x : x\sqrt{3} : 2x\)
• x = Short leg (Opposite \(30^\circ\))
• 2x = Hypotenuse
• \(x\sqrt{3}\) = Long leg (Opposite \(60^\circ\))
Trig Ratios
SOH
Sine is Opposite over Hypotenuse
\(\sin \theta = \frac{O}{H}\)
CAH
Cosine is Adjacent over Hypotenuse
\(\cos \theta = \frac{A}{H}\)
TOA
Tangent is Opposite over Adjacent
\(\tan \theta = \frac{O}{A}\)
Finding Angles (Inverse)
To find a missing angle , use the inverse trigonometric functions on your calculator.
1. \(\sin x = \frac{3}{5}\)
2. \(x = \sin^{-1}(\frac{3}{5})\)
3. \(x \approx 36.87^\circ\)
Calculator Tip!
Make sure your calculator is in DEGREE mode, not Radian mode!
DEG
Real-World Apps
Angle of Elevation
Looking UP from the horizontal line.
Angle of Depression
Looking DOWN from the horizontal line.
Horizontal Sight Line Depression Elevation
