Solving the Triangle Perimeter using Angle Bisector Theorem

The problem asks for the perimeter of a triangle ΔABC where an angle bisector of ∠A meets the side BC at point D. We are given the lengths of two sides (AB and AC) and the length of one segment of the third side (BD).

Given information:

• AB = 9.6 cm
• AC = 11.2 cm
• BD = 4.8 cm
• AD is the angle bisector of ∠A.

We need to find the perimeter of ΔABC, which is AB + AC + BC. To find the perimeter, we first need to find the length of the side BC. The side BC is divided into two segments, BD and DC, by the angle bisector AD. We know BD, so we need to find DC.

Applying the Angle Bisector Theorem

The Angle Bisector Theorem states that in a triangle, the angle bisector of an angle divides the opposite side in the ratio of the other two sides. In ΔABC, since AD is the angle bisector of ∠A, according to the Angle Bisector Theorem:

$$ \frac{AB}{AC} = \frac{BD}{DC} $$

We can substitute the given values into this equation:

$$ \frac{9.6}{11.2} = \frac{4.8}{DC} $$

Now, we can solve for DC using cross-multiplication:

$$ 9.6 \times DC = 11.2 \times 4.8 $$

$$ DC = \frac{11.2 \times 4.8}{9.6} $$

To simplify the calculation, we can observe that 9.6 is exactly twice of 4.8.

$$ DC = \frac{11.2 \times 4.8}{2 \times 4.8} $$

$$ DC = \frac{11.2}{2} $$

$$ DC = 5.6 \text{ cm} $$

So, the length of the segment DC is 5.6 cm.

Calculating the Length of Side BC

The side BC is the sum of the segments BD and DC:

$$ BC = BD + DC $$ $$ BC = 4.8 \text{ cm} + 5.6 \text{ cm} $$ $$ BC = 10.4 \text{ cm} $$

The length of the side BC is 10.4 cm.

Calculating the Perimeter of ΔABC

The perimeter of a triangle is the sum of the lengths of its three sides.

Perimeter of ΔABC = AB + AC + BC

Perimeter = 9.6 cm + 11.2 cm + 10.4 cm

Perimeter = (9.6 + 11.2) + 10.4 cm Perimeter = 20.8 + 10.4 cm Perimeter = 31.2 cm

The perimeter of ΔABC is 31.2 cm.

Let's summarize the lengths of the sides:

Perimeter = 9.6 + 11.2 + 10.4 = 31.2 cm.

Revision Table: Key Concepts

Additional Information: Understanding the Angle Bisector Theorem

The Angle Bisector Theorem is a fundamental concept in geometry that relates the lengths of the sides of a triangle to the lengths of the segments created by an angle bisector. The theorem is useful in various problems involving triangles, especially when dealing with proportions and side lengths.

Consider a triangle PQR. If PS is the angle bisector of ∠P, where S is a point on QR, then the theorem states that the ratio of the length of side PQ to the length of side PR is equal to the ratio of the length of segment QS to the length of segment SR. Mathematically, this is written as:

$$ \frac{PQ}{PR} = \frac{QS}{SR} $$

This theorem has a converse as well: If a point on a side of a triangle divides the side into segments proportional to the other two sides, then the line segment from the opposite vertex to this point is the angle bisector.

The Angle Bisector Theorem is derived from the Sine Rule or by constructing parallel lines and using properties of similar triangles or transversals. It's a powerful tool for solving problems involving side lengths and angle bisectors in triangles.