Download Lesson 10: Obtuse Scalene Triangles - Logo-Math

Obtuse Scalene Triangles
Lesson 10: Obtuse Scalene Triangles
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?
Please review the material in Lesson 7 before completing this lesson.
Example 1: Obtuse Scalene Triangle XYZ
In this example, you will use both the sine function and the Pythagorean theorem (PT) 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, an auxiliary line was
drawn perpendicular from X to segment ZY specifically to form two right triangles (∆XPZ and
∆XPY). 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 here 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 for the altitude will determine the remaining dimensions.
Figure 10.1
Notice that all of the angle measurements are given for triangle XYZ. Since segment XP is
perpendicular to segment ZY, we know that ∠XPZ and ∠XPY are right angles. Thus m(∠PXY) is
63° and m(∠PXZ) is 53° (the sum of the measures of angles in a triangle is 180°). The
calculations used to find the missing side lengths and the Logo code are on the next page (start
at point Z).
KG SHAFER, ©2007 revision date – August 2012
Obtuse Scalene Triangles
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
sin (27°) = 35/c
.454 ≈ 35/c
.454 * c ≈ 35
c ≈ 35/.454
c ≈ 77.1 units
b2 = 58.12 – 352
b2 = 2150.61
b ≈ 46.4 units
d2 = 77.12 – 352
d2 = 4719.41
d ≈ 68.7 units
Completed Figure
Figure 10.2
Finish the problem by creating the logo code and graphic.
Logo Code Template
Logo Code
RT (90 – 37)
FD 58.1
RT (180 – 116)
FD 77.1
RT (180 – 27)
FD (68.7 + 46.4)
RT 53
FD 58.1
RT 64
FD 77.1
RT 153
FD 115.1
Completed Graphic
Figure 10.3
KG SHAFER, ©2007 revision date – August 2012
Obtuse Scalene Triangles
Example 2: Obtuse Scalene Triangle QRS
In this example, I will walk through the steps used to complete calculations for triangle QRS.
Figure 10.4
My decision making process generally depends on the following guidelines:
A. Use the theorem that says the sum of the measures of a triangle equals 180°.
B. If I am given one side length of a triangle, I use the sine function with the measure of the
angle opposite the given side length.
C. If I have two side lengths of the triangle, I use the PT with the two given lengths.
Step One – I see there are two right triangles (because of the auxiliary line drawn from point P
perpendicular to segment QS). Apply guideline A.
m(∠RQT) = 40° and m(∠SRT) = 65°
Step Two – I see the hypotenuse of triangle RTQ measures 120 pixels. Apply guideline B.
sin (50°) = w/120
sin (50°) * 120 = w
92 pixels ≈ w
Step Three – I see two of the three side lengths for triangle RTQ. Apply guideline C.
1202 – 922 = x
77 pixels ≈ x
Step Four – I see the leg of triangle RTS measures 77 pixels. Apply guideline B.
sin (25°) = 77/y
sin (25°) * y = 77
sin (25°) * y = 77/[sin (25°)] 182.2 ≈ y
Step Five – I see two of the three side lengths for triangle RTS. Apply guideline C.
182.22 – 772 = z
165 pixels ≈ z
KG SHAFER, ©2007 revision date – August 2012
Obtuse Scalene Triangles
Logo Code Template
RT 50
FD 120
RT (180-115)
FD 182.2
RT (180-25)
FD (165 + 92)
Logo Code
RT 50
FD 120
RT 65
FD 182.2
RT 155
FD 257
Completed Graphic
LESSON 10 - TASK 1: Write and test the Logo code for the obtuse triangle show here.
Figure 10.6
LESSON 10 - TASK 2: Write and test the Logo Code for the obtuse triangle shown here.
Figure 10.7
KG SHAFER, ©2007 revision date – August 2012
Obtuse Scalene Triangles
Lesson 10 Project
Write and test the Logo code for the shape shown in figure
10.8. Use the grid to approximate segment lengths.
Hint – there are matching obtuse isosceles triangles.
Figure 10.8
KG SHAFER, ©2007 revision date – August 2012