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Sagot :
To determine which statement is true about an isosceles right triangle, let's consider the properties of such a triangle.
### Properties of an Isosceles Right Triangle
An isosceles right triangle has two equal legs and a right angle (90 degrees) between them. Let's denote the length of each leg by [tex]\( a \)[/tex].
### Applying the Pythagorean Theorem
The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For an isosceles right triangle, the hypotenuse [tex]\( c \)[/tex] can be calculated as:
[tex]\[ c^2 = a^2 + a^2 \][/tex]
[tex]\[ c^2 = 2a^2 \][/tex]
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = a\sqrt{2} \][/tex]
Here, [tex]\( c \)[/tex] represents the length of the hypotenuse and is equal to [tex]\( a\sqrt{2} \)[/tex].
### Evaluating the Statements
Now, let's evaluate each statement:
A. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
- This would mean [tex]\( a = \sqrt{2} \cdot c \)[/tex].
- However, we know [tex]\( c = a\sqrt{2} \)[/tex], and solving for [tex]\( a \)[/tex] gives [tex]\( a = \frac{c}{\sqrt{2}} \)[/tex], not [tex]\( a = \sqrt{2} \cdot c \)[/tex]. This statement is false.
B. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
- This would mean [tex]\( c = \sqrt{3} \cdot a \)[/tex].
- However, we derived that [tex]\( c = a\sqrt{2} \)[/tex]. This statement is false.
C. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
- This would mean [tex]\( a = \sqrt{3} \cdot c \)[/tex].
- As mentioned earlier, [tex]\( a = \frac{c}{\sqrt{2}} \)[/tex], not [tex]\( a = \sqrt{3} \cdot c \)[/tex]. This statement is false.
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
- This would mean [tex]\( c = \sqrt{2} \cdot a \)[/tex].
- This matches our derived formula [tex]\( c = a\sqrt{2} \)[/tex].
### Conclusion
The correct and true statement about an isosceles right triangle is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
### Properties of an Isosceles Right Triangle
An isosceles right triangle has two equal legs and a right angle (90 degrees) between them. Let's denote the length of each leg by [tex]\( a \)[/tex].
### Applying the Pythagorean Theorem
The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For an isosceles right triangle, the hypotenuse [tex]\( c \)[/tex] can be calculated as:
[tex]\[ c^2 = a^2 + a^2 \][/tex]
[tex]\[ c^2 = 2a^2 \][/tex]
[tex]\[ c = \sqrt{2a^2} \][/tex]
[tex]\[ c = a\sqrt{2} \][/tex]
Here, [tex]\( c \)[/tex] represents the length of the hypotenuse and is equal to [tex]\( a\sqrt{2} \)[/tex].
### Evaluating the Statements
Now, let's evaluate each statement:
A. Each leg is [tex]\(\sqrt{2}\)[/tex] times as long as the hypotenuse.
- This would mean [tex]\( a = \sqrt{2} \cdot c \)[/tex].
- However, we know [tex]\( c = a\sqrt{2} \)[/tex], and solving for [tex]\( a \)[/tex] gives [tex]\( a = \frac{c}{\sqrt{2}} \)[/tex], not [tex]\( a = \sqrt{2} \cdot c \)[/tex]. This statement is false.
B. The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as either leg.
- This would mean [tex]\( c = \sqrt{3} \cdot a \)[/tex].
- However, we derived that [tex]\( c = a\sqrt{2} \)[/tex]. This statement is false.
C. Each leg is [tex]\(\sqrt{3}\)[/tex] times as long as the hypotenuse.
- This would mean [tex]\( a = \sqrt{3} \cdot c \)[/tex].
- As mentioned earlier, [tex]\( a = \frac{c}{\sqrt{2}} \)[/tex], not [tex]\( a = \sqrt{3} \cdot c \)[/tex]. This statement is false.
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
- This would mean [tex]\( c = \sqrt{2} \cdot a \)[/tex].
- This matches our derived formula [tex]\( c = a\sqrt{2} \)[/tex].
### Conclusion
The correct and true statement about an isosceles right triangle is:
D. The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times as long as either leg.
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