QQuestion
Question
What type of special right triangle can an isosceles triangle be?
A. 30°- 60°- 90°
B. 45°- 45°- 90°
C. Both a 30°- 60°- 90° and a 45°- 45°- 90°
D. None
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Answer
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Step 1:: Recall the definition of an isosceles triangle.
An isosceles triangle is a triangle with at least two sides of equal length.
Step 2:: Examine the properties of special right triangles.
Special right triangles are right triangles with specific angle measurements that allow for easy calculation of side lengths. The two most common special right triangles are the 30°- 60°- 90° triangle and the 45°- 45°- 90° triangle.
Step 3:: Analyze the 30°- 60°- 90° triangle.
In a 30°- 60°- 90° triangle, the sides are in the ratio of x : x\sqrt{3} : 2x, where x is the length of the shorter leg. This ratio is derived from the sine and cosine of 30° and 60°.
Step 4:: Analyze the 45°- 45°- 90° triangle.
In a 45°- 45°- 90° triangle, the sides are in the ratio of x : x : x\sqrt{2}, where x is the length of either leg. This ratio is derived from the sine and cosine of 45°.
Step 5:: Compare isosceles triangles with special right triangles.
An isosceles triangle has at least two sides of equal length. In a special right triangle, the legs may or may not be of equal length. However, the hypotenuse is always longer than either leg.
Step 6:: Evaluate the options provided.
Option A, 30°- 60°- 90°, has legs of different lengths. Option B, 45°- 45°- 90°, has legs of equal length. Option C suggests that an isosceles triangle can be both types, which is not possible since the leg lengths are different in the 30°- 60°- 90° triangle. Option D, None, is the correct answer because an isosceles triangle cannot be exclusively classified as either a 30°- 60°- 90° or a 45°- 45°- 90° triangle.
Final Answer
An isosceles triangle cannot be exclusively classified as a 30°- 60°- 90° or a 45°- 45°- 90° triangle.
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