Solving for the Area of Triangle ACD in a Right-Angled Triangle

In this article, we will explore the problem of finding the area of Triangle ACD within a right-angled triangle ABC where AB 12, AC 5, and #916A 90°. The point D is located on the side BC such that the perimeters of triangles ACD and AOD are equal.

Given Data and Calculations

We start with the given right-angled triangle ABC with an angle at A being 90°, AB 12 units, and AC 5 units.

Step 1: Calculate the Hypotenuse BC

BC is the hypotenuse of the right-angled triangle, which can be calculated using the Pythagorean theorem:

$$BC sqrt{AB^2 AC^2} sqrt{12^2 5^2} sqrt{144 25} sqrt{169} 13$$

Step 2: Set Up the Perimeter Equations

The perimeters of triangles ACD and AOD are given to be equal. Let BD x and CD 13 - x.

Perimeters of Triangles ACD and AOD

The perimeter of ACD is:

$$5 AD CD 5 AD (13 - x)$$

The perimeter of AOD is:

$$12 AD x$$

Since the perimeters are equal:

$$5 AD 13 - x 12 AD x$$

Combining like terms and solving for x:

$$18 - x 12 x$$

Subtract 12 from both sides:

$$6 - x x$$

Add x to both sides:

$$6 2x$$

Divide by 2:

$$x 3$$

Hence, BD 3 units and CD 10 units.

Step 3: Calculate the Area of Triangle ACD

The area of Triangle ACD can be calculated using the formula for the area of a triangle, where the base is AC 5 units, the height can be derived from the sine of the angle, and the sine of C is given as sin C 12/13.

The area of Triangle ACD is:

$$Area_{ACD} frac{1}{2} times AC times CD times sin C frac{1}{2} times 5 times 10 times frac{12}{13} frac{300}{13} approx 23.07692 units^2$$

Conclusion and Verification

Thus, the area of Triangle ACD is approximately 23.07692 square units.

Further Exploration

Alternatively, the problem can be solved using more complex methods such as Heron's formula, but the steps above are straightforward and efficient. Applying these steps will help you accurately calculate the area in similar problems involving right-angled triangles.