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On right triangle upper A upper B upper C, the altitude of vertex upper C is drawn as a solid line that extends from upper C to the hypotenuse, upper A upper B. The intersection of the altitude and upper A upper B is labeled upper O.

Given is the right triangle ABC with altitude CO . Which one of the following statements is true and contributes to the proof of the Pythagorean Theorem?

(1 point)
Responses

△ABC∼△BOC
triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△ABC∼△AOC
triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△AOC∼△BOC
triangle upper A upper B upper C congruent to triangle upper A upper C upper O

△AOC∼△COB
triangle upper A upper B upper C congruent to triangle upper A upper C upper O

7 answers

To determine which statement is true and contributes to the proof of the Pythagorean Theorem, let's analyze the relationships between the triangles formed when the altitude \( CO \) is drawn from vertex \( C \) to the hypotenuse \( AB \).

  1. The triangles \( \triangle ABC \) and \( \triangle AOC \) share angle \( A \).
  2. The triangles \( \triangle ABC \) and \( \triangle BOC \) share angle \( B \).
  3. The triangle \( \triangle AOC \) is right-angled at \( O \), and so is triangle \( \triangle BOC \).

Using the property of similar triangles, we can establish the following relationships:

  • Triangle \( \triangle AOC \) is similar to triangle \( \triangle ABC \) since they share angle \( A \) and both have a right angle. Hence \( \triangle ABC \sim \triangle AOC \).
  • Triangle \( \triangle BOC \) is similar to triangle \( \triangle ABC \) since they share angle \( B \) and both have a right angle. Hence \( \triangle ABC \sim \triangle BOC \).

Based on these observations, the correct relationships contributing to the proof of the Pythagorean Theorem are:

  • \( \triangle AOC \sim \triangle ABC \)
  • \( \triangle BOC \sim \triangle ABC \)

Therefore, the statement that is true is: △AOC∼△ABC.

Among the responses provided, the correct response is: △AOC∼△COB. (This can be represented as similar triangles contributing to the relationship defined by the Pythagorean Theorem, assuring that \( a^2 + b^2 = c^2 \), where \( c \) is hypotenuse.)

If the only options present are indeed the ones listed in your original question, we must pick the closest valid statement from the responses provided, considering the proof methodology illustrated.

In conclusion, the most relevant response from the options given that highlights the similarity relationships contributing to the proof is: △AOC∼△COB, triangle A B C congruent to triangle A C O.

On right triangle upper A upper C upper B, the altitude of vertex upper A is drawn as a solid line that extends from upper A to the hypotenuse, upper B upper C. The intersection of the altitude and upper B upper C is labeled upper D.
Use the steps from the proof of the Pythagorean Theorem to find x.

(1 point)
Responses

x=12
x equals 12

x=169144
x equals Start Fraction 169 over 144 End Fraction

x=16912
x equals Start Fraction 169 over 12 End Fraction

x=1698

To find the value of \( x \) in the context you've provided involving a right triangle \( ACB \) with an altitude \( AD \) drawn to the hypotenuse \( BC \), we need to use the specific relationships and proportions created by similar triangles.

  1. Let's denote:

    • \( a = AC \) (one leg of the triangle)
    • \( b = AB \) (the other leg)
    • \( c = BC \) (the hypotenuse)
    • \( AD \) is the altitude drawn from \( A \) to \( BC \).

  2. In such a configuration, we have:

    • \( AD \) splits \( BC \) into two segments: \( BD \) and \( DC \).
    • By the Pythagorean Theorem, we know that \( AC^2 + AB^2 = BC^2 \).

  3. The relationship established by the altitude \( AD \) gives us similar triangles:

    • \( \triangle ADB \sim \triangle ADC \sim \triangle ABC \).
    • From the similarity of triangles, we can derive the relationships: \[ \frac{AD}{AC} = \frac{BD}{AB} \quad \text{and} \quad \frac{AD}{AB} = \frac{DC}{AC}. \]

  4. The area relationship can show that: \[ AD^2 = BD \cdot DC, \] and we can express \( x \) based on the values derived from the lengths of the legs and hypotenuse.

Without specific values for \( AC \), \( AB \), or the length of \( AD \), it’s challenging to pull an exact value for \( x \).

However, you have provided four potential values for \( x \). If the problem or context gave numerical lengths for the triangle's sides or hypotenuse, you would substitute those into the relationships derived from the similarities and set up an equation to solve for \( x \).

Since you are seeking a numerical answer from the options:

  1. x = 12
  2. x = \(\frac{169}{144}\)
  3. x = \(\frac{169}{12}\)
  4. x = 1698

If any values are given or context is provided that suggests certain dimensions or relationships, please provide that, and I can help calculate \( x \) further. Otherwise, it's necessary to base the answer on computations or interpretations potentially relevant to the triangle's characteristics.

If \( x \) refers to a known segment or dimension that could fit a common Pythagorean triple or a derived consequence from the relationships noted above, then if we assume values like \( c^2 = 169 \) could yield \( x = 12 \) directly through straightforward deduction.

Without those specifics, it’s difficult to indicate the best response outright for calculation-driven decision-making. If you have more context or dimensions, please share them for a precise analysis.

A line, x, divides a right triangle from the vertex opposite the hypotenuse to the hypotenuse. The line x forms a second right angle, marked by a small square, where it meets the hypotenuse, and it divides the hypotenuse into 5 and 12.
Use the geometric mean to solve for the value of x.

(1 point)
Responses

2.65
2.65

60
60

7.75
7.75

17

To solve for the value of \( x \) using the geometric mean in the context you’ve provided, let’s summarize the scenario.

Given a right triangle where:

  • The hypotenuse is divided into two segments: \( 5 \) and \( 12 \).
  • The point where the line intersects the hypotenuse (forming a right angle) creates two smaller segments, \( BD \) and \( DC \).

According to the properties of similar triangles in right triangles with altitude to the hypotenuse, we know the following relationship holds:

\[ x^2 = BD \cdot DC, \]

where:

  • \( BD = 5 \)
  • \( DC = 12 \)

Thus, we can express the equation as:

\[ x^2 = 5 \cdot 12. \]

Now, calculate \( 5 \cdot 12 \):

\[ x^2 = 60. \]

To find \( x \), take the square root of both sides:

\[ x = \sqrt{60}. \]

Now simplifying \( \sqrt{60} \):

\[ x = \sqrt{4 \cdot 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15}. \]

Now let's compute \( 2\sqrt{15} \) numerically. We want to evaluate \( \sqrt{15} \):

\[ \sqrt{15} \approx 3.872 (approx) \Rightarrow 2 \times 3.872 \approx 7.744, \]

which is approximately \( 7.75 \).

Thus, the value of \( x \) is approximately:

\[ \boxed{7.75}. \]

So the correct response from your options is:

7.75.

(1 point)
Responses

15.5
15.5

18
18

24
24

21

It appears that you've provided a new set of response options without context. Let's clarify what we're addressing so we can find the correct answer.

From your previous question regarding the calculation using the geometric mean in a triangle, we found that the geometric mean \( x \) (derived from the segments of the hypotenuse) resulted in approximately \( 7.75 \). If this is part of a larger problem where you are now considering new measurements or values that lead to a different calculation, please provide additional context or specifics.

If we interpret that you're looking to find a similar relationship involving the hypotenuse or another right triangle situation, we should perform a new calculation or explore a new scenario.

If this relates to the same triangle or angle measurements but doesn't involve the geometric mean directly and the values of \( 15.5, 18, 24, \) and \( 21 \) seem reasonable in the context of triangle side calculations, please confirm the specifics (like the dimensions of the triangle or what exactly you're trying to solve), and I will assist you accordingly.

If the intention is to apply a similar approach or if there were corrections to previous dimensions not just involving the results from before, clarification will be crucial in answering the new designations you've posed.

If you can provide more information on how these new choices relate to the previous triangle or the conditions for calculating \( x \), it will help me provide a more accurate answer and guidance.