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Chapter 1: Problem 28

True or False: A right triangle can also be isosceles.

Short Answer

Expert verified
True: A right triangle can also be isosceles, as demonstrated by the case when the two legs are equal in length, and the Pythagorean theorem (\(a^2 + b^2 = c^2\)) still holds.

Step by step solution

01

Understand the Properties of a Right Triangle

A right triangle has one angle measuring 90 degrees. Let's call the two sides that form the right angle as legs (a and b) and the longest side opposite to the right angle as the hypotenuse (c). The relationship between these sides is given by the Pythagorean theorem: \(a^2 + b^2 = c^2\). Step 2:
02

Understand the Properties of an Isosceles Triangle

An isosceles triangle is a triangle with at least two sides of equal length. In our case, we will evaluate whether two of the sides a, b, and c in a right triangle can be of equal length. Step 3:
03

Check If a Right Triangle Can Have Two Equal Sides and Still Satisfy the Pythagorean Theorem

Let's consider the case when the two legs (a and b) are equal in length. That is \(a = b\), and we will see if the triangle still satisfies the Pythagorean theorem: \((a)^2 + (a)^2 = c^2\) \(2a^2 = c^2\) Step 4:
04

Conclusion

Since we showed in Step 3 that a right triangle can have two legs of equal length and still satisfy the Pythagorean theorem, we can conclude that a right triangle can also be isosceles. So, the statement is True.

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