The length of the one leg of the triangle is \(\boxed{\bf 9\sqrt{2}\text{\bf\ cm}}\).
Further explanation:
Given:
The length of hypotenuse of a triangle is \(18\text{ cm}\).
Calculation:
If one angle of the triangle is \(90^{\circ}\) then the triangle is said to be a right angle triangle.
From attached Figure 1 it can be seen that \(AC\) is the hypotenuse, \(BC\) is the base side and \(AB\) is perpendicular in a triangle \(\triangle\text{ ABC}\).
In \(\triangle\text{ ABC}\), the base length can be calculated as follows:
\(\text{cos}45^{\circ}=\dfrac{BC}{AC}\) …… (1)
It is given that the value of \(AC\) is \(18\text{ cm}\). The value of \(\text{cos}45^{\circ}\) is \(\frac{1}{\sqrt{2}}\).
Now, substitute \(AC=18\) and \(\text{cos}45^{\circ}=\frac{1}{\sqrt{2}}\) in equation (1) to obtain the value of length of \(BC\).
\(\begin{aligned}\dfrac{1}{\sqrt{2}}&=\dfrac{BC}{18}\\BC&=\dfrac{18}{\sqrt{2}}\\&=\dfrac{18}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\\&=\dfrac{18\sqrt{2}}{2}\\&=9\sqrt{2}\end{aligned}\)
Therefore, the length of perpendicular side \(BC\) is \(9\sqrt{2}\text{ cm}\).
In \(\triangle\text{ ABC}\), the perpendicular length can be calculated as,
\(\text{sin}45^{\circ}=\dfrac{AB}{AC}\) …… (2)
Now, substitute \(AC=18\) and \(\text{sin}45^{\circ}=\frac{1}{\sqrt{2}}\) in equation (2) to obtain the length of side \(AB\) as follows:
\(\begin{aligned}\dfrac{1}{\sqrt{2}}&=\dfrac{AB}{18}\\AB&=\dfrac{18}{\sqrt{2}}\\&=\dfrac{18}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\\&=\dfrac{18\sqrt{2}}{2}\\&=9\sqrt{2}\end{aligned}\)
Therefore, the length of perpendicular side \(AB\) is \(\boxed{\bf 9\sqrt{2}\text{\bf cm}}\).
Learn more:
1. Learn more about graph: https://brainly.com/question/2334270
2. Learn more about problem on triangle https://brainly.com/question/7437053
3. Learn more about problem on angle https://brainly.com/question/3717797
Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Triangles
Keywords: Triangle, perpendicular, base, hypotenuse, cos, sin, cos45, sin45, right angles triangle, figure, length, 45-45-90, 18cm, 9square root 2.
