Exploring the Geometry of a Right-Angled Triangle with Angle Bisector and Perpendicular

In the field of geometry, understanding the relationships between different parts of a right-angled triangle can provide valuable insights into solving complex problems. This article will delve into the solution of a specific problem related to the angle bisector and a perpendicular in a right-angled triangle, offering a detailed step-by-step approach.

Problem Statement

Given a right-angled triangle ABC, where angle BAC 90°, the bisector AD intersects BC. If DE is perpendicular to AC, with AB 4 cm and AC 6 cm, we are to find the value of (frac{12}{DE}) in cm.

Step-by-Step Solution

Step 1: Find the Length of Side BC

We begin by using the Pythagorean theorem in triangle ABC:

[ BC^2 AB^2 AC^2 4^2 6^2 16 36 52 ] [ BC sqrt{52} 2sqrt{13} , text{cm} ]

Step 2: Find the Length of the Angle Bisector AD

Using the Angle Bisector Theorem, the lengths BD and DC are proportional to the adjacent sides AB and AC:

[ frac{BD}{DC} frac{AB}{AC} frac{4}{6} frac{2}{3} ]

Let BD 2x and DC 3x. Then:

[ BD cdot DC BC implies 2x cdot 3x 2sqrt{13} implies 6x^2 2sqrt{13} implies x^2 frac{2sqrt{13}}{6} frac{sqrt{13}}{3} ] [ x sqrt{frac{sqrt{13}}{3}} frac{sqrt[4]{169}}{sqrt{3}} frac{sqrt{13}}{sqrt{3}} frac{2sqrt{13}}{5} ]

Hence:

[ BD 2x frac{4sqrt{13}}{5}, quad DC 3x frac{6sqrt{13}}{5} ]

Step 3: Find the Length of DE

Now, DE is the length of the perpendicular from D to AC. We can find DE using the formula for the length of the angle bisector AD:

[ AD frac{2 cdot AB cdot AC}{AB AC} cdot cosleft(frac{angle BAC}{2}right) ]

Since (angle BAC 90°), we have (angle BAD angle CAD 45°). Therefore, (cos 45° frac{1}{sqrt{2}}):

[ AD frac{2 cdot 4 cdot 6}{4 6} cdot frac{1}{sqrt{2}} frac{48}{10} cdot frac{1}{sqrt{2}} frac{24}{5sqrt{2}} frac{12sqrt{2}}{5} ]

To find DE, we use the area of triangle ABC in two ways:

[ text{Area of } triangle ABC frac{1}{2} cdot AB cdot AC frac{1}{2} cdot 4 cdot 6 12 , text{cm}^2 ] [ text{Area of } triangle ABC frac{1}{2} cdot AC cdot DE implies 12 frac{1}{2} cdot 6 cdot DE implies 12 3DE implies DE 4 , text{cm} ]

Step 4: Calculate (frac{12}{DE})

Finally, we calculate (frac{12}{DE}):

[ frac{12}{DE} frac{12}{4} 3 ]

Thus, the final answer is:

[ boxed{3} ]