Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Definition Properties Construct Pythagorean Theorem Altitude Theorem

After viewing the video, looking over the pictures, and reading the lesson, you will be able to:

- Identify and define a right triangle
- Understand the identifying property of right triangles
- Prove the Pythagorean theorem
- Prove the right triangle altitude theorem

All triangles have interior angles adding to $180\xb0$. When one of those interior angles measures $90\xb0$, it is a **right angle** and the triangle is a **right triangle**. In drawing right triangles, the interior $90\xb0$ angle is indicated with a little square □ in the vertex.

The term "right" triangle may mislead you to think "left" or "wrong" triangles exist; they do not. "Right" refers to the Latin word *rectus*, meaning "upright."

We already know the square vertex of the right triangle is a right angle. Opposite it is the triangle's **hypotenuse**, the longest of the three sides, usually labeled $c$.

The other two angles in a right triangle add to $90\xb0$; they are complementary. The **sides** opposite the **complementary angles** are the triangle's legs and are usually labeled $a$ and $b$.

A right triangle must have one interior angle of exactly $90\xb0$. It can be scalene or isosceles but never equilateral.

Use two uncooked spaghetti strands to make your own right triangle. Leave one alone; break the other strand into two unequal lengths. Place the two short strands $a$ and $b$ so they meet at two endpoints and form a $90\xb0$ angle. Laying the third strand $c$ down to intersect the two endpoints of $a$ and $b$ creates a right triangle.

You can make a more accurate right triangle by using graph paper and a straightedge. Draw a line segment (of any desired length) along the graph paper's printed lines. Follow the lines to make a second line segment exactly $90\xb0$ to your first line segment, of any desired length. If you connect the two endpoints of those line segments, you have a right triangle.

Geometry uses symbols as shorthand. Here are important ones to know:

∼ means "similar"

$\angle $ means "angle"

$\u25b3$ means "triangle"

Greek mathematician Pythagoras gets the credit, but other civilizations knew about this theorem. The **Pythagorean Theorem** describes the relationship between the lengths of legs $a$ and $b$ of any right triangle to the length of hypotenuse $c$:

The sum of the squares of legs $a$ and $b$ are equal to the square of hypotenuse $c$, or

${a}^{2}+{b}^{2}={c}^{2}$

Thousands of proofs of this theorem exist, including one by U.S. president James Garfield (before he became president). One proof is easy to make with graph paper, a straightedge, pencil, and scissors.

Construct $\u25b3ABC$ with legs $a$ and $b$ to the left and bottom and hypotenuse $c$ at the top right. Leg $a$ is opposite $\angle A$, leg $b$ is opposite $\angle B$, and hypotenuse $c$ is opposite right angle $C$.

Let length $a=3$, $b=4$, and hypotenuse $c=5$.

Construct a square using leg $a$ as the right side of the square. It will be 9 square units (${a}^{2}$). Construct a square using leg $b$ as the top side of its square, so it is 16 square units (${b}^{2}$). Cut out another 5 x 5 square and line it up with hypotenuse $c$, so the square is ${c}^{2}$.

Think: what is 9 square units + 16 square units? It is 25 square units, the area of ${c}^{2}$.

$area=a\times a={a}^{2}$

$area=b\times b={b}^{2}$

$area=c\times c={c}^{2}$

${a}^{2}+{b}^{2}={c}^{2}$

Learn how to use the Pythagorean Theorem to calculate the length of one side of a right triangle.

The right triangle altitude theorem tells us that the altitude of a right triangle drawn to the hypotenuse $c$ forms two similar right triangles that are also similar to the original right triangle.

Construct $\u25b3ABC$ so that hypotenuse $c$ is horizontal and opposite right angle $C$, meaning legs $a$ and $b$ are intersecting above $c$ to form the right angle $C$. This puts $\angle A$ to the bottom left, and $\angle B$ to the bottom right.

Construct an altitude (or height) $h$ from the interior right angle $C$ to hypotenuse $c$ (so it is perpendicular to $c$).

This altitude $h$ creates two smaller triangles inside our original triangle. The altitude divided $\angle C$, and also created two right angles where it intersected hypotenuse $c$.

Call the point where the altitude $h$ touches hypotenuse $c$ point $D$. You now have two triangles, $\u25b3ACD$ and $\u25b3BCD$. Each of these triangles is similar to the other triangle, and both are similar to the original triangle.

You can prove this by seeing that new triangle's $\angle ADC$ = original triangle's $\angle ACB$, while new triangle's $\angle CAD$ = original triangle's $\angle CAB$.

This means two angles of $\u25b3ADC$ and $\u25b3ABC$ are similar, making the triangles themselves similar (by the Angle-Angle postulate or AA postulate):

$\u25b3ADC$ ∼ $\u25b3ABC$.

Go through the figure again, concentrating on the larger, new triangle $BCD$. Here $\angle BDC$ = $\angle ACB$, and $\angle DBC$ = $\angle ABC$, so again, (by the AA postulate):

$\u25b3BCD$ ∼ $\u25b3ABC$.

Since each of the two smaller triangles are similar to the larger triangle, they are similar to each other. So:

$\u25b3ABC$ ∼ $\u25b3BCD$ ∼ $\u25b3ADC$.

After going through the videos, reading the lesson and examining the pictures, you now know how to identify a right triangle (by its interior right angle), what its identifying property is (it has one interior right angle).

You also know what the Pythagorean Theorem is (${a}^{2}+{b}^{2}={c}^{2}$) and how to prove it, and what the right triangle altitude theorem is (the altitude of a right triangle drawn to the hypotenuse $c$ forms two similar right triangles that are also similar to the original right triangle) and how to prove it.

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