Question

1. solve for a and b. a = __ b = __

Explanation:

Step1: Identify the triangle type

This is a 45 - 45- 90 right - triangle. In a 45 - 45- 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$, where the legs are of equal length and the hypotenuse $c$ is related to the leg length $x$ by $c = x\sqrt{2}$.

Step2: Set up the equation for the hypotenuse

Let the length of the legs be $a$ and $b$. Since $a = b$ (because it's a 45 - 45- 90 triangle) and the hypotenuse $c=\sqrt{12}$. We know that $c = a\sqrt{2}$ (or $c = b\sqrt{2}$). So, $\sqrt{12}=a\sqrt{2}$.

Step3: Solve for $a$ and $b$

First, simplify $\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$. Then, from $\sqrt{12}=a\sqrt{2}$, we can solve for $a$ by cross - multiplying: $a=\frac{\sqrt{12}}{\sqrt{2}}=\frac{2\sqrt{3}}{\sqrt{2}}$. Rationalize the denominator: $a=\frac{2\sqrt{3}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{2\sqrt{6}}{2}=\sqrt{6}$. Since $a = b$ in a 45 - 45- 90 triangle, $b=\sqrt{6}$.

Answer:

$a = \sqrt{6}$
$b=\sqrt{6}$