Example Question - calculating hypotenuse length
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Calculating the Length of the Hypotenuse of a Right-Angled Triangle
The image displays a right-angled triangle with the two legs measuring 60 meters and 80 meters, and the length of the hypotenuse labeled as "c." To find the length of the hypotenuse, we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The Pythagorean theorem is expressed as: \[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Given the side lengths of 60 meters and 80 meters, we can plug them into the equation: \[ c^2 = 60^2 + 80^2 \] \[ c^2 = 3600 + 6400 \] \[ c^2 = 10000 \] To find \( c \), we take the square root of both sides of the equation: \[ \sqrt{c^2} = \sqrt{10000} \] \[ c = 100 \] Thus, the length of the hypotenuse is 100 meters. There is no need to round to the nearest tenth, since we have an exact value.
Calculating the Length of a Right Triangle's Hypotenuse
The triangle shown in the image is a right triangle, and we are asked to find the length of the hypotenuse (labeled "c"). To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula for the Pythagorean theorem is: a² + b² = c² From the image, we have side lengths: a = 60 meters (the horizontal side) b = 80 meters (the vertical side) Now we substitute these values into the formula: (60 m)² + (80 m)² = c² 3600 m² + 6400 m² = c² 10000 m² = c² Taking the square root of both sides to solve for c, we get: c = √10000 m² c = 100 m Therefore, the length of the hypotenuse c is 100 meters. There is no need to round because the result is an exact whole number.
Calculating hypotenuse length using geometric mean relationships
In the given right triangle \( ABC \), with altitude \( BD \) drawn to the hypotenuse \( AC \), and \( BD = 6 \) and \( DC = 12 \), we can find the length of \( AC \) by using the geometric mean relationships that exist in a right triangle with an altitude drawn to its hypotenuse. The length of \( AC \) is the sum of \( AD \) and \( DC \). We know \( DC = 12 \), but we need to find \( AD \) before we can find \( AC \). The altitude \( BD \) is the geometric mean of the two segments of the hypotenuse. So we have \( BD^2 = AD \cdot DC \). Given that \( BD = 6 \) and \( DC = 12 \), we plug these values into the equation: \( 6^2 = AD \cdot 12 \) \( 36 = AD \cdot 12 \) Now, solve for \( AD \): \( AD = \frac{36}{12} = 3 \) Now that we know \( AD \) is 3 and \( DC \) is 12, we can find the length of \( AC \) by adding these two lengths: \( AC = AD + DC = 3 + 12 = 15 \) So, the length of the hypotenuse \( AC \) is 15 units.
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