Guided Practice

A right triangle is a triangle with one 90° angle. The longest side of a right triangle is always opposite the right angle and is called the hypotenuse (high-PAW-tuh-noose). The other two sides are called the legs.

The hypotenuse length of Triangle PQR can be written as c or as PR or RP.

Press Reset on the diagram.

Each blue segment has a length of **a**. Each red segment has a length of **b**. So, the area of the entire square diagram is ( + )^{2}.

Each side of the white square has a length of **c**. So, the area of the white square is ^{2}.

When you press Play, you'll see that the area of the white square is broken up into two white squares.

The area of one of these squares is ^{2}, and the area of the other is equal to ^{2}.

The total area of the entire square diagram did not change. So, c^{2} = ^{2} + ^{2}.

Press Reset. Then click and drag on a corner of the white square to change its position. Then press Play again. Do you come to the same conclusion?

The Pythagorean Theorem says that, in a right triangle, the sum of the squares of each shorter side (a^{2} + b^{2}) is equal to the square of the longest side (c^{2}).

Or, ^{2} + ^{2} = c^{2}.

The approximate length of the hypotenuse is given. Write the hypotenuse length for Triangle DEF using the capital vertex label letters (for example, AB). Then complete the formula with the numeric side lengths.

The hypotenuse of Triangle DEF is .

^{2} + ^{2} ≈ 5.39^{2}

The approximate length of the hypotenuse is given. Write the hypotenuse length using the vertex labels. Then complete the formula with the numeric side lengths.

The hypotenuse of Triangle WXZ is .

^{2} + ^{2} ≈ 9.43^{2}

The approximate lengths of the legs are given. Write the hypotenuse length using the vertex labels. Then complete the formula with the numeric side length.

The hypotenuse of Triangle MNP is .

7.07^{2} + 7.07^{2} ≈ ^{2}

Determine the integer length of the hypotenuse, given the leg measurements.

a = 9 units, b = 40 units

c = units

Determine the integer length of the hypotenuse, given the leg measurements.

a = 5 units, b = 12 units

c = units

Determine the integer length of the hypotenuse, given the leg measurements.

a = 6 units, b = 8 units

c = units

Determine the integer length of the hypotenuse, given the leg measurements.

a = 12 units, b = 16 units

c = units

Determine the integer length of the hypotenuse, given the leg measurements.

a = 28 units, b = 45 units

c = units

Determine the integer length of the hypotenuse, given the leg measurements.

a = 20 units, b = 21 units

c = units

The Pythagorean Theorem also works in three dimensions. The x- and y-axes are at right angles to each other, and any line segment in the x-y plane is at a right angle to the z-axis.

It may be difficult to visualize, but the image shows that c^{2} = x^{2} + y^{2} and s^{2} = c^{2} + z^{2}.

So, s^{2} = x^{2} + y^{2} + z^{2}. The square of a hypotenuse in 3 dimensions is equal to x^{2} + y^{2} + z^{2}.

Triangle ADC and Triangle ABC have three pairs of equal corresponding side lengths.

BC =

AC = AC

AB =

This means the triangles are congruent.

The video shows a proof of the Converse of the Pythagorean Theorem.

This theorem says that if a triangle has side lengths such that ^{2} + ^{2} = ^{2}, then it must be a triangle.

Complete the formula by writing side lengths and the words "equals" or "does not equal". Then say whether the triangle "is" or "is not" a right triangle.

^{2} + ^{2} ^{2}

The triangle a right triangle.

Complete the formula by writing side lengths and the words "equals" or "does not equal". Then say whether the triangle "is" or "is not" a right triangle.

^{2} + ^{2} ^{2}

The triangle a right triangle.

Complete the formula by writing side lengths and the words "equals" or "does not equal". Then say whether the triangle "is" or "is not" a right triangle.

^{2} + ^{2} ^{2}

The triangle a right triangle.

^{2} + ^{2} ^{2}

The triangle a right triangle.

^{2} + ^{2} ^{2}

The triangle a right triangle.

Determine the value of c^{2}.

c^{2} = units

Determine the value of c^{2}.

c^{2} = units

Jay walks 3 blocks north and then 4 blocks east to get to the store. If he walks straight back home, how far does Jay walk in all?

3^{2} + 4^{2} = 25

So, c^{2} = 25, which means c = 5. Thus, Jay's walk back home is 5 blocks, and he walks 3 + 4 + 5, or 12, blocks in all.

TV screen sizes are given by their diagonal measure from a top corner to the opposite bottom corner.

What is the approximate size of this TV to the nearest inch?

TV size: in.

The base of a ladder is 14 ft away from an apartment building. The ladder reaches up to a height of 48 ft. How long is the ladder?

The ladder is ft long.

A scout troop hiked 6 miles south and 8 miles west from camp to visit a scenic view. They took the shortest path straight back to camp. How many miles did they hike in all?

They hiked a total of miles.

The side of a cone forms the hypotenuse of a triangle with the radius of the base and the height. What is the height of the cone?

The cone's height is cm.

When three integers a, b, and c satisfy the equation a^{2} + b^{2} = c^{2}, they are called a Pythagorean triple. Enter a true Pythagorean triple.

a =

b =

c =