3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Triangles and Their Properties

Every triangle has one key property: its three interior angles always add up to 180°. This means if you know two angles, you can always find the third by subtracting from 180°.

Example: If two angles of a triangle measure 45° and 90°, the third angle is $180° - 45° - 90° = 45°$.

Triangles are classified by their side lengths and angle measures:

• Equilateral: All three sides are equal length, and all three angles are 60°
• Isosceles: Two sides are equal length, and the angles opposite those equal sides are also equal
• Scalene: All three sides are different lengths, and all three angles are different

Area and Perimeter of Triangles

The area of a triangle uses the formula:

$A = \frac{1}{2}bh$

where $b$ is the length of the base and $h$ is the height (measured perpendicular to the base). The height must form a 90° angle with the base, which is not necessarily the same as one of the triangle's sides.

Example: A triangular sail has a base of 8 ft and a height of 12 ft. Its area is $A = \frac{1}{2}(8)(12) = 48$ square feet.

The perimeter of a triangle is found by adding all three side lengths together. If a triangular garden has sides of 5 m, 7 m, and 9 m, the perimeter is $5 + 7 + 9 = 21$ meters.

Pythagorean Theorem for Side Lengths

The Pythagorean theorem applies only to right triangles (triangles with one 90° angle). It states:

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

where $c$ is the hypotenuse (the longest side, always opposite the right angle) and $a$ and $b$ are the two legs.

To solve for an unknown side:

1. Identify which side is the hypotenuse and which are the legs
2. Substitute the known values into $a^2 + b^2 = c^2$
3. Solve for the unknown variable
4. Take the square root (and use only the positive value, since lengths can't be negative)

Example: A right triangle has legs of 3 and 4. Find the hypotenuse.

$3^2 + 4^2 = c^2$ $9 + 16 = c^2$ $25 = c^2$ $c = 5$

Example: A right triangle has one leg of 5 and a hypotenuse of 13. Find the other leg.

$5^2 + b^2 = 13^2$ $25 + b^2 = 169$ $b^2 = 144$ $b = 12$

A common mistake is forgetting which value goes with $c$. The hypotenuse is always the longest side and always sits across from the right angle. If you're solving for a leg, you subtract: $b^2 = c^2 - a^2$.

Rectangle Properties in Practical Problems

A rectangle has four sides with all angles equal to 90°. Opposite sides are parallel and equal in length, and its two diagonals are equal in length and bisect each other.

The area of a rectangle is:

$A = lw$

where $l$ is the length and $w$ is the width.

The perimeter of a rectangle is:

$P = 2l + 2w$

This can also be written as $P = 2(l + w)$.

Example: A room is 12 ft long and 9 ft wide. The area is $A = (12)(9) = 108$ square feet (useful for buying flooring). The perimeter is $P = 2(12) + 2(9) = 24 + 18 = 42$ feet (useful for buying baseboard trim).

One useful fact: rectangles can share the same area but have very different perimeters. A 4 × 9 rectangle and a 6 × 6 square both have an area of 36 square units, but the 4 × 9 has a perimeter of 26 while the 6 × 6 has a perimeter of 24. The closer a rectangle is to a square, the smaller its perimeter for a given area.

Foundations of Geometric Reasoning

Geometry is the study of shapes, sizes, and positions of figures in space. The work in this section falls under Euclidean geometry, which deals with flat surfaces and straight lines.

A theorem is a mathematical statement that has been proven true based on definitions, axioms, and previously proven statements. The Pythagorean theorem is one of the most well-known theorems in Euclidean geometry, and you'll use it repeatedly in algebra and beyond whenever right triangles appear.