Question

mpm 2d1 - assignment #2 - analytic geometry
communication: thinking
communication:

1. the side lengths of triangle abc are ab = 16 units, bc = 16 units, and ac = 12 units and the slopes of the line - segments are ab = 1, bc = - 1/7, and ac=-1.

a) is triangle abc equilateral, isosceles or scalene? explain.
2
diagram of triangle abc with some hand - written notes
b) is it a right - angled triangle? explain.
yes, because ab is 1 and ac=-1 meaning that ab and ac are negative reciprocals
2

1. the side lengths of quadrilateral abcd are ab = 11 units, bc = 13 units, cd = 11 units, and ad = 13 units. what types of quadrilateral could abcd be? explain.

Explanation:

Step1: Recall triangle - type definitions

An equilateral triangle has all sides equal, an isosceles triangle has two sides equal, and a scalene triangle has all sides different. In triangle ABC, AB = 16 units, BC = 16 units, and AC = 12 units.

Step2: Determine triangle type

Since AB = BC = 16 units and AC = 12 units, triangle ABC has two equal - length sides. So, it is an isosceles triangle.

Step3: Recall right - triangle condition

Two lines with slopes \(m_1\) and \(m_2\) are perpendicular if \(m_1\times m_2=- 1\). The slope of AB is 1 and the slope of AC is - 1.

Step4: Check if it's a right - triangle

Calculate the product of the slopes of AB and AC: \(1\times(-1)=-1\). So, AB and AC are perpendicular, and triangle ABC is a right - angled triangle.

Step5: Recall quadrilateral properties

For a quadrilateral ABCD with AB = 11 units, BC = 13 units, CD = 11 units, and AD = 13 units.

Step6: Determine quadrilateral type

Since AB = CD and BC = AD, the opposite sides of the quadrilateral are equal. A quadrilateral with opposite sides equal is a parallelogram.

Answer:

a) Triangle ABC is isosceles because two of its side lengths (AB and BC) are equal.
b) Triangle ABC is a right - angled triangle because the slopes of AB and AC are negative reciprocals of each other.

1. Quadrilateral ABCD could be a parallelogram because its opposite sides are equal (AB = CD and BC = AD).