Right-Angled Triangle Perimeter Calculation
The problem asks for the perimeter of a right-angled triangle ABC, with the right angle at C. We are given the length of the hypotenuse AB and information about a point P on BC.
Pythagorean Theorem Application
In right-angled triangle ABC:
\(AC^2 + BC^2 = AB^2\)
Given \(AB = 20\) cm:
\(AC^2 + BC^2 = 20^2 = 400 \quad \cdots (1)\)
P is the midpoint of BC, so \(PC = \frac{BC}{2}\).
Consider the right-angled triangle ACP:
\(AC^2 + PC^2 = AP^2\)
Given \(AP = 4\sqrt{13}\) cm:
\(AC^2 + \left(\frac{BC}{2}\right)^2 = (4\sqrt{13})^2\)
\(AC^2 + \frac{BC^2}{4} = 16 \times 13 = 208 \quad \cdots (2)\)
Finding Side Lengths
Subtract equation (2) from equation (1):
\((AC^2 + BC^2) - \left(AC^2 + \frac{BC^2}{4}\right) = 400 - 208\)
\(BC^2 - \frac{BC^2}{4} = 192\)
\(\frac{3}{4} BC^2 = 192\)
\(BC^2 = 192 \times \frac{4}{3} = 64 \times 4 = 256\)
\(BC = \sqrt{256} = 16 \text{ cm}\)
Substitute \(BC^2 = 256\) into equation (1):
\(AC^2 + 256 = 400\)
\(AC^2 = 400 - 256 = 144\)
\(AC = \sqrt{144} = 12 \text{ cm}\)
Perimeter Calculation
The perimeter of triangle ABC is \(AB + BC + AC\).
Perimeter = \(20 \text{ cm} + 16 \text{ cm} + 12 \text{ cm}\)
Perimeter = 48 cm
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