Question

question 6 (1 point)
find the value of x.
27 9
blank 1:
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Explanation:

Step1: Recall Geometric Mean Theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Wait, no, here we have a right triangle with an altitude, and the segment adjacent to \( x \) is 9, and the other segment is 27? Wait, actually, the Geometric Mean (Leg) Theorem: In a right triangle, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So if we let the entire hypotenuse be \( 27 + 9=36 \)? Wait, no, looking at the diagram, the two segments on the hypotenuse are 27 and 9, and \( x \) is a leg of the smaller right triangle. So by the Geometric Mean Theorem (Leg Rule), \( x^2=9\times(27 + 9) \)? Wait, no, wait. Wait, the large triangle is a right triangle, and the altitude is drawn to the hypotenuse? Wait, no, the diagram shows a right triangle with a segment from the right angle to the hypotenuse, creating two smaller right triangles. Wait, actually, the correct theorem is: In a right triangle, when an altitude is drawn to the hypotenuse, then each leg of the original triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, no, the two smaller triangles are similar to the original triangle and to each other. So the triangle with leg \( x \) and base 9 is similar to the triangle with base \( 27 + 9 \) and hypotenuse (the leg of the original triangle). Wait, maybe I got it wrong. Wait, the segment adjacent to \( x \) is 9, and the other segment is 27, so the entire hypotenuse of the large triangle is \( 27+9 = 36 \)? No, wait, the two segments on the base (hypotenuse) are 27 and 9, so the length of the hypotenuse of the large right triangle is \( 27 + 9=36 \)? Wait, no, the large triangle has a right angle at the top, and the altitude is drawn to the hypotenuse (the base) from the right angle, creating two smaller right triangles. Then, by the Geometric Mean Theorem, the length of the leg \( x \) (which is a leg of the smaller right triangle with base 9) is equal to the geometric mean of the hypotenuse segment adjacent to it (9) and the sum of the two segments (27 + 9)? Wait, no, the correct formula is: If in a right triangle, the altitude to the hypotenuse is \( h \), and the two segments of the hypotenuse are \( a \) and \( b \), then \( h^2 = a\times b \), and each leg \( l_1 \) and \( l_2 \) satisfies \( l_1^2=a\times(a + b) \) and \( l_2^2=b\times(a + b) \). Wait, no, let's label the triangle: Let the large right triangle be \( \triangle ABC \) with right angle at \( A \), and \( AD \) is the altitude to hypotenuse \( BC \), where \( BD = 27 \), \( DC = 9 \), so \( BC=36 \). Then, \( AC^2=DC\times BC \), because \( \triangle ADC \sim \triangle BAC \). So \( AC^2=9\times(27 + 9)=9\times36 \). Then \( AC=\sqrt{9\times36}=\sqrt{324}=18 \)? Wait, no, wait, \( x \) is \( AC \)? Wait, the diagram shows \( x \) as the leg of the triangle with base 9. So according to the Geometric Mean Theorem (Leg Rule), the leg of the right triangle ( \( x \)) is the geometric mean of the hypotenuse segment adjacent to it (9) and the entire hypotenuse (27 + 9 = 36). So \( x^2=9\times(27 + 9) \)? Wait, no, \( x^2=9\times(27 + 9) \)? Wait, \( 9\times36 = 324 \), so \( x=\sqrt{324}=18 \)? Wait, no, wait, maybe the other way. Wait, the triangle with base 27 and the triangle with base 9: the ratio of similarity. Wait, the two smaller triangles are similar. So \( \frac{x}{9}=\frac{27 + 9}{x} \)? Wait, no, cross - multiplying, \( x^2=9\times(27 + 9) \)? Wait, no, let's use the geometric mean theorem correctly. In a right triangle, when you draw an altitude from the right angle to the hypotenuse, then \( (leg)^2=(segment\ adjacent\ to\ leg)\times(hypotenuse) \). So if the leg is \( x \), the segment adjacent to \( x \) is 9, and the hypotenuse is \( 27 + 9=36 \), then \( x^2=9\times36 \). Then \( x=\sqrt{9\times36}=\sqrt{9}\times\sqrt{36}=3\times6 = 18 \)? Wait, no, wait, maybe I mixed up the segments. Wait, the two segments on the hypotenuse are 27 and 9, so the length of the hypotenuse is \( 27+9 = 36 \). The leg \( x \) is adjacent to the segment of length 9, so by the geometric mean theorem, \( x^2=9\times(27 + 9) \)? Wait, no, the formula is \( (leg)^2=(segment\ of\ hypotenuse\ adjacent\ to\ leg)\times(hypotenuse) \). So yes, \( x^2=9\times36 \), so \( x = \sqrt{324}=18 \). Wait, but let's check again. The three triangles: the large one, and the two small ones, are all similar. So the triangle with legs \( x \) and 9 is similar to the triangle with legs \( (27 + 9) \) and \( x \)? Wait, no, the triangle with base 9 and leg \( x \) is similar to the triangle with base \( x \) and leg \( 27 \)? Wait, maybe the correct proportion is \( \frac{x}{9}=\frac{27 + 9}{x} \), which is \( x^2=9\times36 \), so \( x = 18 \).

Step2: Calculate \( x \)

We have \( x^2=9\times(27 + 9) \)
First, calculate the sum inside the parentheses: \( 27+9 = 36 \)
Then, multiply: \( 9\times36=324 \)
Take the square root of both sides: \( x=\sqrt{324}=18 \)

Answer:

\( 18 \)