This unit is all about right triangles. Right triangles are triangles with one right angle, making the other two angles acute. Since right triangles have many special properties and patterns, this unit is devoted to the study of those characteristics.
The Pythagorean Theorem relates the side lengths of a right triangle. Prior to using the Pythagorean Theorem, it is important that you can decipher the difference between the hypotenuse and the legs of a triangle. The hypotenuse is located opposite from the right angle and is the longest side of a right triangle. The legs of a right triangle are the two sides not labelled as the hypotenuse. |
Pythagorean Theorem
The sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse
\(leg^2 + leg^2 = hypotenuse^2\)
The sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse
\(leg^2 + leg^2 = hypotenuse^2\)
Example 1a:
Given \(m\angle A = 90^{\circ}\), \(AN = 10\) units, and \(SA = 8\) units, find \(SN\). Solution: \(\begin{align*} (SA)^2 + (AN)^2 &= (SN)^2\\ (8)^2 + (10)^2 &= (SN)^2\\ 64 + 100 &= (SN)^2\\ 164 &= (SN)^2\\ \pm\sqrt{164} &= SN \end{align*}\) Since length is positive we will use the positive value. \(SN = \sqrt{164}=\sqrt{4\cdot41}=\sqrt4\sqrt{41}=2\sqrt{41}\) \(SN\approx12.8\) |
Example 1b:
Given \(m\angle A = 90^{\circ}\), \(JY = 12\) units, and \(JA = 4\) units, find \(AY\). Solution: \(\begin{align*} (JA)^2 + (AY)^2 &= (JY)^2\\ (4)^2 + (AY)^2 &= (12)^2\\ 16 + (AY)^2 &= 144\\ (AY)^2 &= 128\\ AY &= \pm\sqrt{128} \end{align*}\) Since length is positive we will use the positive value. \(AY = \sqrt{128}=\sqrt{64}\cdot{2}=\sqrt{64}\sqrt{2}=8\sqrt{2}\) \(AY\approx11.3\) |
Pythagorean Triples
A Pythagorean Triple consists of three positive integers that satisfy the Pythagorean Theorem. Some of the Pythagorean Triples are listed below.
A Pythagorean Triple consists of three positive integers that satisfy the Pythagorean Theorem. Some of the Pythagorean Triples are listed below.
Converse of the Pythagorean Theorem
Suppose you are given three side lengths of a triangle. You can use the converse of the Pythagorean Theorem to determine you if the triangle is an acute triangle (all three interior angles are acute), a right triangle, or an obtuse triangle (one angle is obtuse and the other two are acute).
Suppose you are given three side lengths of a triangle. You can use the converse of the Pythagorean Theorem to determine you if the triangle is an acute triangle (all three interior angles are acute), a right triangle, or an obtuse triangle (one angle is obtuse and the other two are acute).
Important sidebar: Be careful, when you are given three side lengths, you have to first determine if a triangle can exist from those side lengths. Forget how to determine that?
Example 2a:
With the given side lengths, \(15\) cm, \(25\) cm, \(20\) cm determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Step 1: First , determine the longest side: \(25\)
Step 2: Then, check to see if a triangle can be formed:
\(15 + 20 > 25?\)
\(35 > 25\)
Since the sum of the two smaller side lengths is greater than the third side length, then a triangle CAN be formed!
Step 3: Now, test to see if it’s acute, obtuse, or right:
\(25^2\) ? \(15^2 + 20^2\)
\(625\) = \(225 + 400\)
Answer: This triangle is a RIGHT triangle.
Example 2b:
With the given side lengths, \(17\) in, \(14\) in, \(2\) in determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Step 1: First , determine the longest side: \(17\)
Step 2: Then, check to see if a triangle can be formed:
\(14 + 2 > 17?\)
\(16 \ngtr 17\)
Since the sum of the two smaller side lengths IS NOT greater than the third side length, a triangle CANNOT be formed!
Answer: No triangle can be formed from these side lengths.
Example 2c:
With the given side lengths, \(12\) mi, \(7\) mi, \(7\) mi determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Watch the video for the solution.
Example 2a:
With the given side lengths, \(15\) cm, \(25\) cm, \(20\) cm determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Step 1: First , determine the longest side: \(25\)
Step 2: Then, check to see if a triangle can be formed:
\(15 + 20 > 25?\)
\(35 > 25\)
Since the sum of the two smaller side lengths is greater than the third side length, then a triangle CAN be formed!
Step 3: Now, test to see if it’s acute, obtuse, or right:
\(25^2\) ? \(15^2 + 20^2\)
\(625\) = \(225 + 400\)
Answer: This triangle is a RIGHT triangle.
Example 2b:
With the given side lengths, \(17\) in, \(14\) in, \(2\) in determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Step 1: First , determine the longest side: \(17\)
Step 2: Then, check to see if a triangle can be formed:
\(14 + 2 > 17?\)
\(16 \ngtr 17\)
Since the sum of the two smaller side lengths IS NOT greater than the third side length, a triangle CANNOT be formed!
Answer: No triangle can be formed from these side lengths.
Example 2c:
With the given side lengths, \(12\) mi, \(7\) mi, \(7\) mi determine if there can be no triangle formed or if the triangle formed is acute, obtuse, or right.
Solution:
Watch the video for the solution.
Quick Check
1) With the given side lengths, determine if the triangle is acute, obtuse, right, or no triangle can be formed: \(13\) meters, \(12\) meters, \(7\) meters
2) There is a triangle with side lengths \(22\) ft and \(6\) ft. Assuming that \(22\) ft is the length of the longest side, what could be a length of the third side so that the triangle remains obtuse?
1) With the given side lengths, determine if the triangle is acute, obtuse, right, or no triangle can be formed: \(13\) meters, \(12\) meters, \(7\) meters
2) There is a triangle with side lengths \(22\) ft and \(6\) ft. Assuming that \(22\) ft is the length of the longest side, what could be a length of the third side so that the triangle remains obtuse?