Lesson #10 - Obtuse Scalene Triangles

Initial posting - July, 2011
Revision date - Fall 2017

Please review the material in Lesson 7 before completing this lesson.Purpose: You will use the sine function and the Pythagorean theorem to find missing side measurements in obtuse scalene triangles. Logo will be used to check your work.Essential Questions: How can an obtuse scalene triangle be decomposed into two right triangles? How can the sine function and the Pythagorean theorem be used to create the Logo code for an obtuse scalene triangle?

A Section of the Lesson Follows

Example 1: Obtuse Scalene Triangle XYZ
In this example, you will use both the sine function and the Pythagorean theorem to create the Logo code for an obtuse scalene triangle. Triangle XYZ has angles that measure 116°, 27° and 37°. This is clearly an obtuse scalene triangle. In the diagram below, a perpendicular auxiliary line was drawn perpendicular from X to segment ZY specifically to form two right triangles (∆XPY and ∆XPZ). The purpose of drawing the auxiliary line was to decompose the triangle into two right triangles. A segment drawn from a vertex perpendicular to the opposite side of a triangle is called the altitude.
In this example, I selected the three angle measures and I made segment XP = 35 units. You might wonder why I had to pick a measure for only one of the sides. If you think about it, the angle measures will determine the general shape of the triangle, but there are an infinite number of similar triangles (of different sizes) that will have angles that measure 116°, 27° and 37° degrees. The length of 35 units for the altitude will determine the remaining dimensions.

Notice that all of the angle measurements are given for triangle XYZ. Since segment XP is a perpendicular line, we know that XPZ and XPY are right angles. Thus m(PXZ) is 53° and m(PXY) is 63° (the sum of the measures of angles in a triangle is 180°). The calculations used to find the missing side lengths for right triangle XPZ and then for right triangle XPY.

Computation for triangle XPZ
Computation for triangle XPY
sin(37°) = 35/a
.602 ≈ 35/a
.602 * a ≈ 35
a ≈ 35/.602
a ≈ 58.1 units

b^2 = 58.12 – 352
b^2 = 2150.61
b ≈ 46.4 units
sin (27°) = 35/c
.454 ≈ 35/c
.454 * c ≈ 35
c ≈ 35/.454
c ≈ 77.1 units

d^2 = 77.12 – 352
d^2 = 4719.41
d ≈ 68.7 units

Omitted Section of Lesson #10

LESSON 10 - TASK 2: Write and test the Logo Code for the obtuse triangle shown here: