homework # w22
jmsa - 314 a. pilana
find the missing side lengths. leave your answers as radicals in simplest form.
1)
2)
3)
4)
5)
6)

Question

Accepted Answer

### Problem 1
# Explanation:
## Step1: Identify triangle type (45-45-90)
It's a 45-45-90 triangle, so legs are equal, hypotenuse = leg$\times\sqrt{2}$. One leg is 20, so $x = 20$ (other leg), $y = 20\sqrt{2}$ (hypotenuse).
## Step2: Verify using ratios
In 45-45-90, leg : leg : hypotenuse = $1:1:\sqrt{2}$. So $x = 20$, $y = 20\sqrt{2}$.
# Answer:
$x = 20$, $y = 20\sqrt{2}$

### Problem 2
# Explanation:
## Step1: Identify triangle type (30-60-90)
It's a 30-60-90 triangle. The side opposite 30° is shorter leg, opposite 60° is longer leg, hypotenuse = 2×shorter leg, longer leg = shorter leg$\times\sqrt{3}$. Here, longer leg (opposite 60°) is $6\sqrt{3}$, so shorter leg $b = 6$ (since $6\sqrt{3}=b\sqrt{3}\Rightarrow b = 6$), hypotenuse $a = 12$ (since $a = 2b$).
## Step2: Check ratios
30-60-90 ratios: shorter leg : longer leg : hypotenuse = $1:\sqrt{3}:2$. So $b = 6$, $a = 12$.
# Answer:
$a = 12$, $b = 6$

### Problem 3
# Explanation:
## Step1: Identify triangle type (45-45-90)
It's a 45-45-90 triangle. One leg is $7\sqrt{2}$, so the other leg $y = 7\sqrt{2}$, hypotenuse $x = 7\sqrt{2}\times\sqrt{2}=14$.
## Step2: Use ratios
Leg : leg : hypotenuse = $1:1:\sqrt{2}$. So $y = 7\sqrt{2}$, $x = 14$.
# Answer:
$x = 14$, $y = 7\sqrt{2}$

### Problem 4
# Explanation:
## Step1: Identify triangle type (30-60-90)
It's a 30-60-90 triangle. The side opposite 30° is 17 (shorter leg), so hypotenuse $x = 34$ (since $x = 2\times17$), longer leg $y = 17\sqrt{3}$ (since $y = 17\sqrt{3}$).
## Step2: Check ratios
Shorter leg : longer leg : hypotenuse = $1:\sqrt{3}:2$. So $y = 17\sqrt{3}$, $x = 34$.
# Answer:
$x = 34$, $y = 17\sqrt{3}$

### Problem 5
# Explanation:
## Step1: Identify triangle type (45-45-90)
It's a 45-45-90 triangle. Hypotenuse is $14\sqrt{3}$? Wait, no, wait: Wait, the hypotenuse is $14\sqrt{3}$? Wait, no, in 45-45-90, legs are equal, hypotenuse = leg$\times\sqrt{2}$. Wait, the hypotenuse is given as $14\sqrt{3}$? Wait, no, maybe I misread. Wait, the triangle has a 45° angle, right angle, so legs $x = y$, hypotenuse $14\sqrt{3}$? Wait, no, that can't be. Wait, no, maybe the hypotenuse is $14\sqrt{2}$? Wait, no, the problem says $14\sqrt{3}$. Wait, no, maybe it's a typo, but assuming it's 45-45-90, legs $x = y$, hypotenuse $x\sqrt{2}=14\sqrt{3}$? No, that would not be rational. Wait, no, maybe I misidentified. Wait, no, the triangle has 45° and right angle, so it's isoceles right triangle, so legs equal, hypotenuse = leg$\times\sqrt{2}$. So if hypotenuse is $14\sqrt{3}$, then legs $x = y=\frac{14\sqrt{3}}{\sqrt{2}} = 7\sqrt{6}$. But that seems odd. Wait, maybe the hypotenuse is $14\sqrt{2}$? Then $x = y = 14$. But the problem says $14\sqrt{3}$. Wait, maybe the triangle is 45-45-90, so legs equal, hypotenuse = leg$\times\sqrt{2}$. So $x = y$, and hypotenuse $14\sqrt{3}=x\sqrt{2}\Rightarrow x = y=\frac{14\sqrt{3}}{\sqrt{2}} = 7\sqrt{6}$. But maybe the problem has a typo, but following the ratio:
## Step1: 45-45-90 ratios
Leg : leg : hypotenuse = $1:1:\sqrt{2}$. So $x = y=\frac{14\sqrt{3}}{\sqrt{2}} = 7\sqrt{6}$.
## Step2: Rationalize
$\frac{14\sqrt{3}}{\sqrt{2}}=\frac{14\sqrt{6}}{2}=7\sqrt{6}$. So $x = y = 7\sqrt{6}$.
# Answer:
$x = 7\sqrt{6}$, $y = 7\sqrt{6}$

### Problem 6
# Explanation:
## Step1: Identify triangle type (30-60-90)
It's a 30-60-90 triangle. The side opposite 30° is 19? No, wait, the leg adjacent to 30° is 19? Wait, no, the leg with length 19 is opposite 60°? Wait, no, the right angle, 30° angle, so the side opposite 30° is the shorter leg, opposite 60° is longer leg, hypotenuse is $x$. Wait, the leg with length 19 is the longer leg (opposite 60°), so shorter leg $y$, longer leg $19 = y\sqrt{3}\Rightarrow y=\frac{19}{\sqrt{3}}=\frac{19\sqrt{3}}{3}$, hypotenuse $x = 2y=\frac{38\sqrt{3}}{3}$. Wait, no, wait: in 30-60-90, the side opposite 30° is shorter leg ($y$), opposite 60° is longer leg ($19$), hypotenuse ($x$). So longer leg = shorter leg$\times\sqrt{3}\Rightarrow 19 = y\sqrt{3}\Rightarrow y=\frac{19}{\sqrt{3}}=\frac{19\sqrt{3}}{3}$, hypotenuse $x = 2y=\frac{38\sqrt{3}}{3}$. But wait, the leg with length 19 is adjacent to 30°, so it's the longer leg (opposite 60°). So:
## Step1: 30-60-90 ratios
Shorter leg ($y$) : longer leg ($19$) : hypotenuse ($x$) = $1:\sqrt{3}:2$. So $y=\frac{19}{\sqrt{3}}=\frac{19\sqrt{3}}{3}$, $x=\frac{38\sqrt{3}}{3}$.
## Step2: Rationalize
$y=\frac{19\sqrt{3}}{3}$, $x=\frac{38\sqrt{3}}{3}$.
# Answer:
$x=\frac{38\sqrt{3}}{3}$, $y=\frac{19\sqrt{3}}{3}$

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QUESTION IMAGE

Question

Explanation:

Problem 1

Step1: Identify triangle type (45-45-90)

Step2: Verify using ratios

Step1: Identify triangle type (30-60-90)

Step2: Check ratios

Step1: Identify triangle type (45-45-90)

Step2: Use ratios

Answer:

Problem 2