Question

essential question: what is the relationship among the side lengths of special right triangles? theorem 9.4 45°-45°-90° triangle theorem in a 45°-45°-90° triangle, the hypotenuse is \\(\sqrt{2}\\) times as long as each leg. theorem 9.5 30°-60°-90° triangle theorem in a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is \\(\sqrt{3}\\) times as long as the shorter leg. 9.2 practice find the missing side lengths. leave your answers as radicals in simplest form. 1) triangle with 30°, right angle, sides 10, v, u 2) triangle with 45°, right angle, hypotenuse \\(10\sqrt{2}\\), sides a, b 3) triangle with 45°, right angle, leg 4, sides x, y 4) triangle with 60°, right angle, leg \\(5\sqrt{3}\\), sides x, y 5) triangle with 30°, 60°, right angle, hypotenuse 10, sides x, y 6) triangle with 45°, right angle, side 6, sides u, v

Explanation:

Step1: Identify triangle type for 1)

This is a 30°-60°-90° triangle. The given leg (10) is opposite the 30° angle, so it is the shorter leg.

Step2: Calculate longer leg (v) for 1)

Longer leg = shorter leg $\cdot \sqrt{3}$
$v = 10 \cdot \sqrt{3} = 10\sqrt{3}$

Step3: Calculate hypotenuse (u) for 1)

Hypotenuse = shorter leg $\cdot 2$
$u = 10 \cdot 2 = 20$
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Step1: Identify triangle type for 2)

This is a 45°-45°-90° triangle. The given side is the hypotenuse $10\sqrt{2}$.

Step2: Calculate legs (a, b) for 2)

Leg = $\frac{\text{hypotenuse}}{\sqrt{2}}$
$a = b = \frac{10\sqrt{2}}{\sqrt{2}} = 10$
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Step1: Identify triangle type for 3)

This is a 45°-45°-90° triangle. The given leg is 4, so the other leg $y=4$.

Step2: Calculate hypotenuse (x) for 3)

Hypotenuse = leg $\cdot \sqrt{2}$
$x = 4 \cdot \sqrt{2} = 4\sqrt{2}$
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Step1: Identify triangle type for 4)

This is a 30°-60°-90° triangle. The given longer leg is $5\sqrt{3}$.

Step2: Calculate shorter leg (y) for 4)

Shorter leg = $\frac{\text{longer leg}}{\sqrt{3}}$
$y = \frac{5\sqrt{3}}{\sqrt{3}} = 5$

Step3: Calculate hypotenuse (x) for 4)

Hypotenuse = shorter leg $\cdot 2$
$x = 5 \cdot 2 = 10$
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Step1: Identify triangle type for 5)

This is a 30°-60°-90° triangle. The given hypotenuse is 10.

Step2: Calculate shorter leg (y) for 5)

Shorter leg = $\frac{\text{hypotenuse}}{2}$
$y = \frac{10}{2} = 5$

Step3: Calculate longer leg (x) for 5)

Longer leg = shorter leg $\cdot \sqrt{3}$
$x = 5 \cdot \sqrt{3} = 5\sqrt{3}$
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Step1: Identify triangle type for 6)

This is a 45°-45°-90° triangle. The given leg is 6, so the other leg $v=6$.

Step2: Calculate hypotenuse (u) for 6)

Hypotenuse = leg $\cdot \sqrt{2}$
$u = 6 \cdot \sqrt{2} = 6\sqrt{2}$

Answer:

1. $v=10\sqrt{3}$, $u=20$
2. $a=10$, $b=10$
3. $x=4\sqrt{2}$, $y=4$
4. $x=10$, $y=5$
5. $x=5\sqrt{3}$, $y=5$
6. $u=6\sqrt{2}$, $v=6$