To solve the problem of determining the height of the glacier, we need to analyze the given information and use trigonometry. Let's break it down step by step.
Step 1: Understand the Problem
- We are given a right triangle.
- One of the angles in the triangle is \(45^\circ\).
- The hypotenuse of the triangle is 20 units.
- We need to find the height of the glacier, which corresponds to the vertical side (opposite the \(45^\circ\) angle) of the triangle.
Step 2: Identify the Type of Triangle
Since one of the angles is \(45^\circ\) and it is a right triangle, this is a
45-45-90 triangle. In a 45-45-90 triangle:
- The two legs are congruent.
- The hypotenuse is \(\sqrt{2}\) times the length of each leg.
Step 3: Use the Properties of a 45-45-90 Triangle
Let the length of each leg be \(x\). According to the properties of a 45-45-90 triangle:
\[
\text{Hypotenuse} = x\sqrt{2}
\]
We are given that the hypotenuse is 20 units. Therefore:
\[
x\sqrt{2} = 20
\]
Step 4: Solve for \(x\)
To find \(x\), divide both sides of the equation by \(\sqrt{2}\):
\[
x = \frac{20}{\sqrt{2}}
\]
Rationalize the denominator:
\[
x = \frac{20}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2}
\]
Step 5: Determine the Height of the Glacier
In a 45-45-90 triangle, the height (which is one of the legs) is equal to \(x\). Therefore, the height of the glacier is:
\[
10\sqrt{2}
\]
Final Answer:
\[
\boxed{10\sqrt{2}}
\]
Parent Tip: Review the logic above to help your child master the concept of measuring mountains worksheet answers.
