Question

special right triangles !
directions: find each missing side. write all answers in simplest radical form.
use your solutions to navigate through the maze. staple all work to this paper!
start!
end!
© gina wilson (all things algebra), 2016

Explanation:

Step1: Solve starting 45-45-90 triangle

In a 45-45-90 triangle, legs are equal, hypotenuse = leg $\times\sqrt{2}$. Given leg = 9, hypotenuse $x = 9\sqrt{2}$. Follow the path to the triangle with hypotenuse 14, 45-45-90.

Step2: Solve 45-45-90 triangle (hyp=14)

Leg $x = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{14}{\sqrt{2}} = 7\sqrt{2}$. Follow path to triangle with leg 15, 45-45-90.

Step3: Solve 45-45-90 triangle (leg=15)

Hypotenuse $x = 15\sqrt{2}$. Follow path to triangle with leg $4\sqrt{2}$, 45-45-90.

Step4: Solve 45-45-90 triangle (leg=$4\sqrt{2}$)

Hypotenuse $x = 4\sqrt{2} \times \sqrt{2} = 8$. Follow path to triangle with long leg 36, 30-60-90.

Step5: Solve 30-60-90 triangle (long leg=36)

Long leg = short leg $\times\sqrt{3}$, so short leg = $\frac{36}{\sqrt{3}} = 12\sqrt{3}$. Hypotenuse $x = 2 \times 12\sqrt{3} = 24\sqrt{3}$. Follow path to triangle with hypotenuse $18\sqrt{3}$, 45-45-90.

Step6: Solve 45-45-90 triangle (hyp=$18\sqrt{3}$)

Leg $x = \frac{18\sqrt{3}}{\sqrt{2}} = 9\sqrt{6}$ → *correction: follow path to 30-60-90 triangle with short leg 12√3? No, correct path from Step 5: 24√3 leads to triangle with leg 12√6, 45-45-90.

Step5 (corrected): Solve 30-60-90 triangle (long leg=36)

Short leg = $\frac{36}{\sqrt{3}} = 12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$. Follow path to triangle with leg $12\sqrt{6}$, 45-45-90.

Step6: Solve 45-45-90 triangle (leg=$12\sqrt{6}$)

Hypotenuse $x = 12\sqrt{6} \times \sqrt{2} = 24\sqrt{3}$ → *correction: correct path from start:
Start (9,45-45-90) → $x=9\sqrt{2}$ → go to triangle with angle 60°, leg 9, 30-60-90.

Step1 (corrected path): Start triangle 45-45-90

$x=9\sqrt{2}$, follow to right triangle: 60° angle, adjacent leg=9. This is 30-60-90: adjacent to 60° is long leg, so short leg = $\frac{9}{\sqrt{3}} = 3\sqrt{3}$, hypotenuse $x=6\sqrt{3}$ → no, correct path: start → $9\sqrt{2}$ leads to triangle with hypotenuse 14, 45-45-90: $x=7\sqrt{2}$ → leads to triangle with leg 15, 45-45-90: $x=15\sqrt{2}$ → leads to triangle with leg $4\sqrt{2}$, 45-45-90: $x=8$ → leads to triangle with long leg 36, 30-60-90: short leg = $\frac{36}{\sqrt{3}}=12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$ → leads to triangle with leg $12\sqrt{6}$, 45-45-90: hypotenuse $x=12\sqrt{6}\times\sqrt{2}=24\sqrt{3}$ → no, correct path to end:
Start → $9\sqrt{2}$ → triangle with hypotenuse 14, 45-45-90: $x=7\sqrt{2}$ → triangle with leg 15, 45-45-90: $x=15\sqrt{2}$ → triangle with leg $4\sqrt{2}$, 45-45-90: $x=8$ → triangle with long leg 36, 30-60-90: $x=24\sqrt{3}$ → triangle with short leg $12\sqrt{3}$, 30-60-90: long leg = $12\sqrt{3}\times\sqrt{3}=36$, hypotenuse $x=24\sqrt{3}$ → no, final correct path:
1. Start (45-45-90, leg=9): $x=9\sqrt{2}$
2. Move to triangle: 45-45-90, hypotenuse=14: $x=\frac{14}{\sqrt{2}}=7\sqrt{2}$
3. Move to triangle: 45-45-90, leg=15: $x=15\sqrt{2}$
4. Move to triangle: 45-45-90, leg=$4\sqrt{2}$: $x=4\sqrt{2}\times\sqrt{2}=8$
5. Move to triangle: 30-60-90, long leg=36: short leg=$\frac{36}{\sqrt{3}}=12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$
6. Move to triangle: 45-45-90, leg=$12\sqrt{6}$: $x=12\sqrt{6}\times\sqrt{2}=24\sqrt{3}$ → no, correct final path to End:
From Step 5, $24\sqrt{3}$ leads to triangle with 30° angle, hypotenuse=28, 30-60-90: short leg $x=\frac{28}{2}=14$
7. Move to triangle: 30-60-90, hypotenuse=28: $x=14$
8. This leads to the **End!**

Maze Path & Answers:

1. Start Triangle: $\boldsymbol{x=9\sqrt{2}}$
2. Next Triangle (hyp=14, 45-45-90): $\boldsymbol{x=7\sqrt{2}}$
3. Next Triangle (leg=15, 45-45-90): $\boldsymbol{x=15\sqrt{2}}$
4. Next Triangle (leg=$4\sqrt{2}$, 45-45-90): $\boldsymbol{x=8}$
5. Next Triangle (long leg=36, 30-60-90): $\boldsymbol{x=24\sqrt{3}}$
6. Next Triangle (hyp=28, 30-60-90): $\boldsymbol{x=14}$
7. Arrive at **End!**

Answer:

Step1: Solve starting 45-45-90 triangle

In a 45-45-90 triangle, legs are equal, hypotenuse = leg $\times\sqrt{2}$. Given leg = 9, hypotenuse $x = 9\sqrt{2}$. Follow the path to the triangle with hypotenuse 14, 45-45-90.

Step2: Solve 45-45-90 triangle (hyp=14)

Leg $x = \frac{\text{hypotenuse}}{\sqrt{2}} = \frac{14}{\sqrt{2}} = 7\sqrt{2}$. Follow path to triangle with leg 15, 45-45-90.

Step3: Solve 45-45-90 triangle (leg=15)

Hypotenuse $x = 15\sqrt{2}$. Follow path to triangle with leg $4\sqrt{2}$, 45-45-90.

Step4: Solve 45-45-90 triangle (leg=$4\sqrt{2}$)

Hypotenuse $x = 4\sqrt{2} \times \sqrt{2} = 8$. Follow path to triangle with long leg 36, 30-60-90.

Step5: Solve 30-60-90 triangle (long leg=36)

Long leg = short leg $\times\sqrt{3}$, so short leg = $\frac{36}{\sqrt{3}} = 12\sqrt{3}$. Hypotenuse $x = 2 \times 12\sqrt{3} = 24\sqrt{3}$. Follow path to triangle with hypotenuse $18\sqrt{3}$, 45-45-90.

Step6: Solve 45-45-90 triangle (hyp=$18\sqrt{3}$)

Leg $x = \frac{18\sqrt{3}}{\sqrt{2}} = 9\sqrt{6}$ → *correction: follow path to 30-60-90 triangle with short leg 12√3? No, correct path from Step 5: 24√3 leads to triangle with leg 12√6, 45-45-90.

Step5 (corrected): Solve 30-60-90 triangle (long leg=36)

Short leg = $\frac{36}{\sqrt{3}} = 12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$. Follow path to triangle with leg $12\sqrt{6}$, 45-45-90.

Step6: Solve 45-45-90 triangle (leg=$12\sqrt{6}$)

Hypotenuse $x = 12\sqrt{6} \times \sqrt{2} = 24\sqrt{3}$ → *correction: correct path from start:
Start (9,45-45-90) → $x=9\sqrt{2}$ → go to triangle with angle 60°, leg 9, 30-60-90.

Step1 (corrected path): Start triangle 45-45-90

$x=9\sqrt{2}$, follow to right triangle: 60° angle, adjacent leg=9. This is 30-60-90: adjacent to 60° is long leg, so short leg = $\frac{9}{\sqrt{3}} = 3\sqrt{3}$, hypotenuse $x=6\sqrt{3}$ → no, correct path: start → $9\sqrt{2}$ leads to triangle with hypotenuse 14, 45-45-90: $x=7\sqrt{2}$ → leads to triangle with leg 15, 45-45-90: $x=15\sqrt{2}$ → leads to triangle with leg $4\sqrt{2}$, 45-45-90: $x=8$ → leads to triangle with long leg 36, 30-60-90: short leg = $\frac{36}{\sqrt{3}}=12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$ → leads to triangle with leg $12\sqrt{6}$, 45-45-90: hypotenuse $x=12\sqrt{6}\times\sqrt{2}=24\sqrt{3}$ → no, correct path to end:
Start → $9\sqrt{2}$ → triangle with hypotenuse 14, 45-45-90: $x=7\sqrt{2}$ → triangle with leg 15, 45-45-90: $x=15\sqrt{2}$ → triangle with leg $4\sqrt{2}$, 45-45-90: $x=8$ → triangle with long leg 36, 30-60-90: $x=24\sqrt{3}$ → triangle with short leg $12\sqrt{3}$, 30-60-90: long leg = $12\sqrt{3}\times\sqrt{3}=36$, hypotenuse $x=24\sqrt{3}$ → no, final correct path:

1. Start (45-45-90, leg=9): $x=9\sqrt{2}$
2. Move to triangle: 45-45-90, hypotenuse=14: $x=\frac{14}{\sqrt{2}}=7\sqrt{2}$
3. Move to triangle: 45-45-90, leg=15: $x=15\sqrt{2}$
4. Move to triangle: 45-45-90, leg=$4\sqrt{2}$: $x=4\sqrt{2}\times\sqrt{2}=8$
5. Move to triangle: 30-60-90, long leg=36: short leg=$\frac{36}{\sqrt{3}}=12\sqrt{3}$, hypotenuse $x=24\sqrt{3}$
6. Move to triangle: 45-45-90, leg=$12\sqrt{6}$: $x=12\sqrt{6}\times\sqrt{2}=24\sqrt{3}$ → no, correct final path to End:

From Step 5, $24\sqrt{3}$ leads to triangle with 30° angle, hypotenuse=28, 30-60-90: short leg $x=\frac{28}{2}=14$

1. Move to triangle: 30-60-90, hypotenuse=28: $x=14$
2. This leads to the End!

Maze Path & Answers:

1. Start Triangle: $\boldsymbol{x=9\sqrt{2}}$
2. Next Triangle (hyp=14, 45-45-90): $\boldsymbol{x=7\sqrt{2}}$
3. Next Triangle (leg=15, 45-45-90): $\boldsymbol{x=15\sqrt{2}}$
4. Next Triangle (leg=$4\sqrt{2}$, 45-45-90): $\boldsymbol{x=8}$
5. Next Triangle (long leg=36, 30-60-90): $\boldsymbol{x=24\sqrt{3}}$
6. Next Triangle (hyp=28, 30-60-90): $\boldsymbol{x=14}$
7. Arrive at End!