Introduction to Altitudes in Right-Angled Triangles
" "Understanding the properties of altitudes in right-angled triangles can provide valuable insights into the geometric relationships and algebraic representations of line equations. This article delves into the calculations and explanations behind the altitudes, focusing on a specific example where two altitudes are known and the third is to be determined.
" "" "Altitudes in Right-Angled Triangles
" "A right-angled triangle is a triangle with one angle of 90 degrees. In such a triangle, the perpendicular sides serve as two of the three altitudes. Given the equation of the line ( frac{x}{a} frac{y}{b} 1 ), the intercepts on the x-axis and y-axis are represented by (a) and (b) respectively. These intercepts are also the lengths of the legs of the triangle, providing a direct relationship between the equation of the line and the geometry of the triangle.
" "" "Example Calculation
" "The problem at hand involves a right-angled triangle with two given altitudes of lengths 5 and 12. One of the altitudes is the perpendicular distance from the origin to the line 12x - 5y 60. To find (a) and (b), the intercepts on the x-axis and y-axis, we proceed as follows:
" "" "Determining the Intercepts
" "To determine the x-intercept, set (y 0), yielding (12x 60), and solving for (x) gives (x 5). For the y-intercept, set (x 0), resulting in (5y 60), and solving for (y) gives (y 12). Thus, the intercepts are at points (A(5, 0)) and (B(0, 12)).
" "" "Calculating the Hypotenuse
" "Using the Pythagorean theorem, the hypotenuse (c) is calculated as follows:
" "[c sqrt{5^2 12^2} sqrt{25 144} sqrt{169} 13.]
" "The area of the triangle can be calculated using the two legs, which are also altitudes, as follows:
" "[text{Area} frac{1}{2} times 5 times 12 30.]
" "" "Finding the Third Altitude
" "The third altitude, which is the perpendicular distance from the origin to the line, can be found using the formula:
" "[h frac{text{Area} times 2}{text{Hypotenuse}} frac{30 times 2}{13} frac{60}{13} approx 4.615.]
" "In terms of a mixed number, the altitude (h 4 frac{8}{13}).
" "" "Sum of the Altitudes
" "The sum of the three altitudes is:
" "[5 12 4 frac{8}{13} 21 frac{8}{13}.]
" "" "Generalization of the Problem
" "The concept can be generalized for a right-angled triangle with side lengths (a), (b), and hypotenuse (c). The two altitudes are (a) and (b), and the third altitude (h) to the hypotenuse is given by:
" "[h frac{ab}{c}.]
" "The sum of the altitudes is:
" "[ab frac{ab}{c} frac{ac bc ac}{c} frac{2ac bc}{c} frac{ac bc ac}{c} frac{2ac bc - bc}{c} frac{ac bc ab}{c}.]
" "For a specific example, if (a 21) and (b frac{8}{13}), the sum of the altitudes simplifies as shown above.
" "" "Conclusion
" "This exploration of altitudes in right-angled triangles not only aids in understanding geometric and algebraic relationships but also provides a practical application in problem-solving scenarios. By knowing two altitudes, the third can be found using the properties of the triangle and the intercept form of a line equation.