Question

1. using the pythagorean theorem, what is the length of the missing side in a right triangle if one leg is 8 cm and the hypotenuse is 17 cm?

a) 15 cm
b) 11 cm
c) 10 cm
d) 9 cm

1. if the legs of a right triangle are 8 cm and 15 cm, what is the length of the hypotenuse?

a) 17 cm
b) 19 cm
c) 20 cm
d) 23 cm

1. find the missing side lengths using special triangle rules

a) (u = \frac{7sqrt{2}}{2}, v=\frac{7sqrt{2}}{2})
b) (u = 7, v = 7)
c) (u=\frac{7sqrt{3}}{2}, v=\frac{7sqrt{3}}{2})
d) (u=\frac{7}{2}, v=\frac{7}{2})

Explanation:

Step1: Recall Pythagorean theorem

The Pythagorean theorem for a right - triangle is $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a,b$ are the legs.

Step2: Solve problem 5

Let one leg $a = 8$ cm and the hypotenuse $c = 17$ cm. We need to find the other leg $b$. Rearranging the Pythagorean theorem gives $b=\sqrt{c^{2}-a^{2}}$. Substitute $a = 8$ and $c = 17$: $b=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=\sqrt{225}=15$ cm.

Step3: Solve problem 6

Let the legs be $a = 8$ cm and $b = 15$ cm. We need to find the hypotenuse $c$. Using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$. Substitute $a = 8$ and $b = 15$: $c=\sqrt{8^{2}+15^{2}}=\sqrt{64 + 225}=\sqrt{289}=17$ cm.

Step4: Solve problem 7

This is a 45 - 45-90 special right - triangle. In a 45 - 45-90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. If the hypotenuse $c = 7\sqrt{2}$, and the legs are $u$ and $v$, and since $u = v$ and $c=\sqrt{2}u=\sqrt{2}v$. Then $u = v=\frac{c}{\sqrt{2}}$. Substitute $c = 7\sqrt{2}$: $u = v=\frac{7\sqrt{2}}{\sqrt{2}}=7$.

Answer:

1. a) 15 cm
2. a) 17 cm
3. b) $u = 7$, $v = 7$