Example Question - right triangle altitude
Here are examples of questions we've helped users solve.
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.
Calculating Altitude of Right Triangle
The problem presents a right triangle ABC with the right angle at B and an altitude BD drawn to the hypotenuse AC. The lengths given are AD = 12 and DC = 16. To find the length of BD in simplest radical form, we're going to make use of similar triangles. The altitude of a right triangle creates two smaller triangles (ABD and CBD) which are similar to each other and also similar to the original triangle ABC. By the properties of similar triangles, the ratios of corresponding sides are equal. So for triangles ABD and ABC, the following ratio holds: AD/AB = AB/AC Given that AD is 12 and AC (the entire hypotenuse) is 12 + 16 = 28, we can say: AB/28 = 12/AB Solving for AB, we get the equation: AB^2 = 12 * 28 AB^2 = 336 AB = √336 AB = √(16*21) AB = 4√21 (since √16 = 4) Now, let us determine the length of BD. The similar triangles ABD and ABC also give us: BD/AB = AB/AC Filling in the values we've found: BD/4√21 = 4√21/28 Cross-multiplying gives: BD = (4√21 * 4√21) / 28 BD = (16 * 21) / 28 BD = 16 * 3/4 (because 21/28 simplifies to 3/4) BD = 12 So the length of BD, which is the altitude of the right triangle ABC, is 12.
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