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

New Concept

This lesson applies your algebra skills to geometry. You'll use formulas for triangles and rectangles, including the Pythagorean Theorem ($a^2 + b^2 = c^2$), to set up and solve equations for unknown lengths, angles, perimeters, and areas.

What’s next

This is just the beginning. Next, you'll work through interactive examples and tackle practice cards to master solving for sides, angles, and areas.

Property

HOW TO: Solve Geometry Applications.
Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Label what we are looking for by choosing a variable to represent it.
Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer by substituting it back into the equation solved in step 5 and by making sure it makes sense in the context of the problem.
Step 7. Answer the question with a complete sentence.

Examples

• To find a missing angle in a triangle, draw the triangle, label the known angles, assign a variable $x$ to the unknown angle, write the equation $A+B+x=180$, and solve for $x$.

• To find a rectangle's width given its area and length, you would draw the rectangle, label the length and area, use $W$ for width, write the formula $A=L \cdot W$, substitute the values, and solve for $W$.

Property

For $\triangle ABC$
Angle measures:
$m\angle A + m\angle B + m\angle C = 180$
The sum of the measures of the angles of a triangle is $180^\circ$.

Perimeter:
$P = a+b+c$
The perimeter is the sum of the lengths of the sides of the triangle.

Area:
$A = \frac{1}{2}bh$, $b$ = base, $h$ = height
The area of a triangle is one-half the base times the height.

Property

In any right triangle, $a^2 + b^2 = c^2$.
where a and b are the lengths of the legs, c is the length of the hypotenuse.
A right triangle has one $90^\circ$ angle. The side of the triangle opposite the $90^\circ$ angle is called the hypotenuse and each of the other sides are called legs.

Examples

• In a right triangle with legs $a=9$ and $b=12$, the hypotenuse $c$ is found using $9^2 + 12^2 = c^2$, which simplifies to $81 + 144 = 225$. Thus, $c = \sqrt{225} = 15$.

• If a right triangle has a hypotenuse $c=17$ and one leg $a=8$, the other leg $b$ is found using $8^2 + b^2 = 17^2$. This gives $64 + b^2 = 289$, so $b^2 = 225$ and $b=15$.

Property

• Rectangles have four sides and four right ($90^\circ$) angles.
• The lengths of opposite sides are equal.
• The perimeter of a rectangle is the sum of twice the length and twice the width.

• The area of a rectangle is the product of the length and the width.

Examples

• The perimeter of a rectangle with a length of 15 meters and a width of 8 meters is $P = 2(15) + 2(8) = 30 + 16 = 46$ meters.

• The area of a rectangle with a length of 12 feet and a width of 7 feet is $A = 12 \cdot 7 = 84$ square feet.