To find the length of \( AD \) in the right triangle \( ABC \), where \( CD \) is the altitude from the right angle \( C \) to the hypotenuse \( AB \), we need to apply both the Pythagorean theorem and properties of similar triangles.
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Identifying the Information: Let's assume:
- The lengths of the two legs \( AC = 4 \) and \( BC = 3 \).
- So, the length of hypotenuse \( AB \) can be calculated using the Pythagorean theorem: \[ AB = \sqrt{AC^2 + BC^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]
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Using the Area: The area \( A \) of triangle \( ABC \) can be found using the legs \( AC \) and \( BC \): \[ A = \frac{1}{2} \times AC \times BC = \frac{1}{2} \times 4 \times 3 = 6. \]
The area can also be expressed in terms of the hypotenuse \( AB \) and the altitude \( CD \): \[ A = \frac{1}{2} \times AB \times CD = \frac{1}{2} \times 5 \times CD. \] Equating the two expressions for the area: \[ 6 = \frac{1}{2} \times 5 \times CD, \] \[ 12 = 5 \times CD \implies CD = \frac{12}{5} = 2.4. \]
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Using Similar Triangles: The triangles \( ACD \) and \( BCD \) are similar to triangle \( ABC \).
- Let \( AD = x \) and \( DB = y \).
- Since \( AD + DB = AB \), we have: \[ x + y = 5. \]
By the properties of similar triangles: \[ \frac{AC}{AB} = \frac{CD}{BC} \implies \frac{4}{5} = \frac{CD}{3} \implies 4 \cdot 3 = 5 \cdot CD \implies CD = \frac{12}{5} = 2.4. \] Hence, using the relation from similar triangles: \[ \frac{AD}{AB} = \frac{AC}{AB} \implies \frac{x}{5} = \frac{4}{5} \implies x = 4. \] Similarly: \[ \frac{DB}{AB} = \frac{BC}{AB} \implies \frac{y}{5} = \frac{3}{5} \implies y = 3. \]
Thus, we find: \[ AD = 4. \] So, the length of \( AD \) is \( 4.00 \) when rounded to two decimal places.
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