Question

which statement about these triangles is true? triangles b and e are obtuse. triangles c and d are isosceles. triangles d and e are scalene. triangles a and b are acute.

Explanation:

Step1: Recall triangle properties

An acute triangle has all angles less than \(90^{\circ}\). An obtuse triangle has one angle greater than \(90^{\circ}\). An isosceles triangle has at least two equal - length sides. A scalene triangle has all sides of different lengths.

Step2: Analyze each option

• Option 1 (Triangles B and E are obtuse):
• For triangle B: All angles are less than \(90^{\circ}\) (by visual inspection of the grid - based triangle, using the fact that in a grid, right - angled triangles can be identified by side - length ratios related to the grid squares, and non - right - angled acute angles can be inferred from the shape).
• For triangle E: One angle is greater than \(90^{\circ}\) (by visual inspection of the side - length relationships on the grid. If we consider the side lengths formed by the grid squares, using the Pythagorean theorem conceptually \(a^{2}+b^{2}
• Option 2 (Triangles C and D are isosceles):
• For triangle C: All sides have different lengths (by counting the number of grid - square units for each side. Let the length of a grid - square side be \(1\). If we assume the vertices of the triangle lie on grid points, and count the horizontal/vertical or diagonal (using the distance formula \(d = \sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)) side lengths.
• For triangle D: All sides have different lengths (using the same grid - based side - length counting and distance - formula - like reasoning as for triangle C). So this option is false.
• Option 3 (Triangles D and E are scalene):
• For triangle D: All sides have different lengths (as per grid - based side - length analysis).
• For triangle E: All sides have different lengths (using the grid - based side - length counting and distance - formula - like approach). But we need to check other properties. However, we can also check triangle A and B.
• Option 4 (Triangles A and B are acute):
• For triangle A: All angles are less than \(90^{\circ}\) (by visual inspection of the grid - based triangle. If we consider the side lengths formed by the grid squares and assume the vertices lie on grid points, using the fact that for a triangle with side lengths \(a,b,c\) (counted on the grid) and using the law of cosines \( \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\), if \(b^{2}+c^{2}>a^{2}\) for all sides \(a,b,c\), the angles are acute).
• For triangle B: All angles are less than \(90^{\circ}\) (using the same grid - based side - length and angle - property analysis as for triangle A).

Answer:

Triangles A and B are acute.