Question

right triangle abc is isosceles and point m is the midpoint of the hypotenuse. what is true about triangle amb? it is congruent to triangle abc. it is an obtuse triangle. it is an isosceles right triangle. it is a scalene triangle.

Explanation:

1. Analyze triangle \(ABC\): Since \(ABC\) is an isosceles right triangle, \(\angle B = 90^\circ\), and \(AB = BC\). The hypotenuse is \(AC\), and \(M\) is the midpoint of \(AC\). In a right - angled triangle, the median to the hypotenuse is half the hypotenuse, so \(AM=BM = CM\).
2. Analyze triangle \(AMB\):

• First, check the angles. \(\angle ABC = 90^\circ\), and since \(AM = BM\), \(\triangle AMB\) is isosceles. Also, \(\angle AMB\): In \(\triangle ABC\), \(\angle A=\angle C = 45^\circ\). Since \(AM = BM\), \(\angle A=\angle ABM = 45^\circ\), so \(\angle AMB=180^\circ-(45^\circ + 45^\circ)=90^\circ\).
• Now, check the options:
• Option 1: \(\triangle AMB\) is not congruent to \(\triangle ABC\) because the size of \(\triangle AMB\) is half (in terms of side - length relationships) of \(\triangle ABC\) (e.g., \(AB\) is a leg of \(\triangle ABC\) and \(AM\) is half of the hypotenuse of \(\triangle ABC\), and their angle - side relationships are different).
• Option 2: \(\triangle AMB\) has a right angle (\(\angle AMB = 90^\circ\)), so it is not obtuse (an obtuse triangle has one angle greater than \(90^\circ\)).
• Option 3: Since \(AM = BM\) (so it is isosceles) and \(\angle AMB=90^\circ\) (so it is a right triangle), \(\triangle AMB\) is an isosceles right triangle.
• Option 4: A scalene triangle has all sides of different lengths, but \(AM = BM\), so it is not scalene.

Answer:

It is an isosceles right triangle.