\(120\text{ cm}^2\)
Triangle Geometry: Angle Bisector Theorem Application
This problem requires applying the Angle Bisector Theorem to a triangle ABC where the angle bisector of \(\angle \text{BAC}\) divides the opposite side BC.
Angle Bisector Theorem Explanation
The Angle Bisector Theorem states that if a line bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle. For triangle ABC, with AD bisecting \(\angle \text{BAC}\):
\(\frac{AB}{AC} = \frac{BD}{DC}\)
Applying the Theorem to Triangle ABC
Given:
- AB = 18 cm
- AC = 15 cm
- BC = 22 cm
Using the Angle Bisector Theorem:
\(\frac{18}{15} = \frac{BD}{DC}\)
Simplify the ratio:
\(\frac{6}{5} = \frac{BD}{DC}\)
This means the ratio \(BD:DC\) is \(6:5\). Since \(BD + DC = BC = 22\) cm, we can find the lengths of BD and DC.
Calculating Segment Lengths
The total ratio parts are \(6 + 5 = 11\).
Calculate BD:
\(BD = \left(\frac{6}{11}\right) \times BC = \left(\frac{6}{11}\right) \times 22 \text{ cm} = 6 \times 2 \text{ cm} = 12 \text{ cm}\)
Calculate DC:
\(DC = \left(\frac{5}{11}\right) \times BC = \left(\frac{5}{11}\right) \times 22 \text{ cm} = 5 \times 2 \text{ cm} = 10 \text{ cm}\)
We can check that \(BD + DC = 12 + 10 = 22\) cm, which is correct.
Final Calculation: Product of Segments
The question asks for the value of (BD × DC).
\(BD \times DC = 12 \text{ cm} \times 10 \text{ cm} = 120 \text{ cm}^2\)
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Important Questions from Geometry
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In ΔABC, ∠A = 90°, AD ┴ BC and AD = BD = 2 cm. The length of CD is
How many lines of symmetry does a rectangle have?
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Which of the following is/are the geometric figures with the line of symmetry?
I. Rectangle
II. Isosceles triangle
