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Wize High School Grade 10 Math Textbook > Analytic Geometry: Geometric Properties

Classifying Triangles

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Classifying Triangles

Every triangle has three sides and three angles.

Notes about Side Length

  • If there are 3 different side lengths, it is a scalene triangle.
  • If there are 2 equal side lengths and 1 different one, it is an isosceles triangle.
  • If there are 3 equal side lengths, it is an equilateral triangle.

Notes about Slopes & Angles

  • The sum of the interior angles in any triangle is 180°180\degree
  • A right triangle has one angle which is exactly 90o90^o (right angle)
  • If the slope of two line segments are negative reciprocal (inverse) of each other, then they are perpendicular (90°90\degree)
  • An obtuse triangle has one angle which is greater than 90o90^o (obtuse angle)
  • An acute triangle has all three angles less than 90o90^o (acute angles)

How to classify a triangle?

  1. Calculate the side lengths to see if the triangle is scalene, isosceles, or equilateral
  2. If the triangle is scalene or isosceles, determine if there's a right angle
*In this course, we do not need to determine if a triangle is obtuse or acute.

Practice: Classifying Triangles

Information about some of the side lengths and slopes of 4 triangles are given below. Classify each triangle.

Triangle 1.

Select all that apply.
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Example: Classifying Triangles

John is building a coffee table where the table top is in the shape of a triangle. Here's the blueprint of the table drawn in the Cartesian Plane. Classify the triangle.

Side AB

x1y1A:(4,3)\begin{array}{ccccc} &x_1&&y_1\\ A:(&4&,&3&) \end{array} and x2y2B:(5,5)\begin{array}{ccccc} &x_2&&y_2\\ B:(&5&,&5&) \end{array}

Side length:
AB=(x2−x1)2+(y2−y1)2=(5−4)2+(5−3)2=(1)2+(2)2=1+4=5\begin{array}{rcl} AB&=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[0.5em] &=&\sqrt{(5-4)^2+(5-3)^2}\\[0.5em] &=&\sqrt{(1)^2+(2)^2}\\[0.5em] &=&\sqrt{1+4}\\[0.5em] &=&\sqrt{5} \end{array}

Slope:
mAB=y2−y1x2−x1=5−35−4=21=2\begin{array}{rcl} m_{AB}&=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] &=&\dfrac{5-3}{5-4}\\[1em] &=&\dfrac{2}{1}\\[1em] &=&2 \end{array}

Side BC

x1y1B:(5,5)\begin{array}{ccccc} &x_1&&y_1\\ B:(&5&,&5&) \end{array} and x2y2C:(6,2)\begin{array}{ccccc} &x_2&&y_2\\ C:(&6&,&2&) \end{array}

Side length:
BC=(x2−x1)2+(y2−y1)2=(6−5)2+(2−5)2=(1)2+(−3)2=1+9=10\begin{array}{rcl} BC&=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[0.5em] &=&\sqrt{(6-5)^2+(2-5)^2}\\[0.5em] &=&\sqrt{(1)^2+(-3)^2}\\[0.5em] &=&\sqrt{1+9}\\[0.5em] &=&\sqrt{10} \end{array}

Slope:
mBC=y2−y1x2−x1=2−56−5=−31=−3\begin{array}{rcl} m_{BC}&=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] &=&\dfrac{2-5}{6-5}\\[1em] &=&\dfrac{-3}{1}\\[1em] &=&-3 \end{array}

Side CA

x1y1C:(6,2)\begin{array}{ccccc} &x_1&&y_1\\ C:(&6&,&2&) \end{array} and x2y2A:(4,3)\begin{array}{ccccc} &x_2&&y_2\\ A:(&4&,&3&) \end{array}

Side length:
CA=(x2−x1)2+(y2−y1)2=(4−6)2+(3−2)2=(−2)2+(1)2=4+1=5\begin{array}{rcl} CA&=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[0.5em] &=&\sqrt{(4-6)^2+(3-2)^2}\\[0.5em] &=&\sqrt{(-2)^2+(1)^2}\\[0.5em] &=&\sqrt{4+1}\\[0.5em] &=&\sqrt{5} \end{array}

Slope:
mCA=y2−y1x2−x1=3−24−6=1−2=−12\begin{array}{rcl} m_{CA}&=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] &=&\dfrac{3-2}{4-6}\\[1em] &=&\dfrac{1}{-2}\\[1em] &=&-\dfrac{1}{2} \end{array}

Since two of the sides lengths AB and CA are the same, this is an isosceles triangle.

Since the slopes of AB and CA are the negative reciprocal of each other, ∠BAC=90°\angle BAC=90\degree.

Therefore, △ABC\triangle ABC is a right isosceles triangle..

Practice: Classifying Triangles

Classify triangle △ABC\triangle ABC that has vertices A(5,3)A(5,3), B(6,5)B(6,5) and C(10,2)C(10,2).

Select all that apply.
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Practice: Classifying Triangles

A triangle is given by the vertices P(1,−2)P(1,-2), Q(3,1)Q(3,1), and R(7,−6)R(7,-6).

a) Show that this is a scalene triangle.

b) Show that this is a right angle triangle by using the slopes of the sides.

c) How can you show that this is a right triangle without using slopes?