Lesson #12 - Acute Scalene Triangles

Initial posting - July, 2011
Revision date - Fall 2017

Purpose: This lesson focuses in decomposing acute scalene triangles into right triangles. You will use what you know about the Pythagorean theorem. The 45-45-90 and the 30-60-90 triangles are introduced in this lesson. Knowledge of the sine function is necessary in the problems marked with an asterisk
Essential Question: Can you apply what you have learned about the sum of the measures of a triangle and the Pythagorean theorem to solve complex problems?

A Section of the Lesson Follows

*Example 1 – Acute Scalene Triangle JKM
In this example, you will use both the sine function and the Pythagorean theorem to write the Logo code for an acute scalene triangle (example 2 does not require the sine function). Triangle JKM has three acute angles and an altitude has been drawn from point K to segment JM.

Work for Triangle KTJ, on the left

Work for Triangle KTM, on the right

y^2 = 200^2 + 200^2
y^2 = 40,000 + 40,000
y ≈ SQRT(80,000) ≈ 282.84 pixels

Note that y = 200 * SQRT (2) = 282.84
The leg measure times the SQRT (2)
equals the measure of the hypotenuse.
This is true for any 45-45-90 triangle.
sin(60) = 200 / z
z = 200 / sin(60°)
z ≈ 230.94 pixels

230.94^2 = 200^2 + w^2
230.94^2 - 200^2 = w^2
115.47 units ≈ w
There is a special relationship between the three side lengths of right triangle KTM ("right side" of the acute triangle). See if you can find it. The solution is at the end of this lesson.

Start the Logo code at point J, moving clockwise.
Logo Template
Logo Drawing
RT 45
FD 282.84
RT (180 - 75)
FD 230.94
RT (180 - 60)
FD (115.47 + 200)
Figure 12.2