Question

1. if the two legs of an isosceles, right triangle have lengths of 8 inches each, which of the following is the length of its hypotenuse in inches?

(1) $8\sqrt{3}$
(2) $16\sqrt{3}$
(3) $8\sqrt{2}$
(4) $16\sqrt{2}$

1. a right triangle has a hypotenuse that is 26 cm long and one leg that is 10 cm long. what is the triangles area, in square centimeters?

(1) 120
(2) $120\sqrt{2}$
(3) 240
(4) $240\sqrt{2}$

Explanation:

Step1: Apply Pythagorean theorem (Q2)

Para un triángulo rectángulo isósceles, el teorema de Pitágoras establece que $c^2 = a^2 + b^2$, donde $a = b = 8$.
$$c^2 = 8^2 + 8^2 = 64 + 64 = 128$$

Step2: Calcular la hipotenusa (Q2)

Despejar $c$ tomando la raíz cuadrada:
$$c = \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}$$

Step3: Encontrar el segundo cateto (Q3)

Usar el teorema de Pitágoras: $a^2 = c^2 - b^2$, donde $c=26$, $b=10$.
$$a^2 = 26^2 - 10^2 = 676 - 100 = 576$$
$$a = \sqrt{576} = 24$$

Step4: Calcular el área (Q3)

El área de un triángulo rectángulo es $\frac{1}{2} \times \text{cateto}_1 \times \text{cateto}_2$.
$$\text{Área} = \frac{1}{2} \times 10 \times 24 = 120$$

Answer:

1. (3) $8\sqrt{2}$
2. (1) 120