Question

The altitude of an isosceles triangle drawn to its base is 3 centimeters, and its perimeter is 18 centimeters. Find the length of its base.

Short answer

8 cm

Step-by-step solution

Define the Variables

Let the lengths of the two equal sides be denoted by \( a \), and the length of the base be \( b \).

Use the Perimeter Formula

The perimeter of the triangle is the sum of all its sides: \( 2a + b = 18 \).

Analyze the Altitude

The altitude draws a perpendicular from the vertex opposite the base, dividing the isosceles triangle into two right triangles. Each right triangle has a base of \( \frac{b}{2} \), a height of 3 cm, and hypotenuse \( a \).

Apply the Pythagorean Theorem

In one of these right triangles, apply the Pythagorean Theorem: \[ a^2 = \left( \frac{b}{2} \right)^2 + 3^2 \].

Simplify the Pythagorean Theorem

This simplifies to: \[ a^2 = \frac{b^2}{4} + 9 \].

Solve the System of Equations

From steps 2 and 5, we have the system of equations: \( 2a + b = 18 \) and \( a^2 = \frac{b^2}{4} + 9 \).

Express \( a \) in Terms of \( b \)

From the first equation, express \( a \) as: \( a = \frac{18 - b}{2} \).

Substitute \( a \) in the Pythagorean Theorem

Substitute \( a = \frac{18 - b}{2} \) into the equation \( a^2 = \frac{b^2}{4} + 9 \): \[ \left( \frac{18 - b}{2} \right)^2 = \frac{b^2}{4} + 9 \].

Simplify the Equation

Simplify the equation: \[ \frac{(18 - b)^2}{4} = \frac{b^2}{4} + 9 \]. \[ (18 - b)^2 = b^2 + 36 \].

Expand and Solve for \( b \)

Expand and solve: \[ 324 - 36b + b^2 = b^2 + 36 \]. \[ 324 - 36b = 36 \]. \[ 288 = 36b \]. \[ b = 8 \].

State the Final Answer

The length of the base is 8 cm.

Key concepts

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles.
It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as:
\[ a^2 + b^2 = c^2 \]
In this exercise, we deal with two right triangles formed by drawing an altitude from the vertex opposite the base of an isosceles triangle.
Each of these right triangles shares a hypotenuse of length \( a \) and a height of 3 cm.
The base of each right triangle is half of the isosceles triangle's base (\(\frac{b}{2}\)). Therefore, using the Pythagorean Theorem in this context:\[ a^2 = \left( \frac{b}{2} \right)^2 + 3^2 \].
This equation is essential for establishing the relationship between the sides of the isosceles triangle and solving for the base length.
By substituting values, we find the necessary measurements of the isosceles triangle.

triangle perimeter

The perimeter of a triangle is the sum of the lengths of all its sides.
For our isosceles triangle, which has two equal sides of length \( a \) and a base of length \( b \), the perimeter is calculated as:
\[ 2a + b = 18 \].
This formula allows us to relate the lengths of the sides of the triangle to its perimeter.
Given the perimeter as 18 cm, we can use this equation to express one side in terms of the other.
For instance, we can derive:
\[ a = \frac{18 - b}{2} \].
This step simplifies the task by reducing the number of variables when using the Pythagorean Theorem.
Once we solve for \( b \), we substitute back to find \( a \), ensuring all side lengths correctly add up to the given perimeter.

right triangles

Understanding right triangles is crucial for solving the problem of finding the base length in an isosceles triangle.
When an altitude is drawn to the base, it forms two congruent right triangles within the isosceles triangle.
The altitude splits the base into two equal segments, each of length \(\frac{b}{2}\).
In each of these right triangles, we have:

• A height (leg) of 3 cm
• A base (half of the original base) of \(\frac{b}{2}\)
• A hypotenuse of length \( a \) (one of the equal sides of the isosceles triangle)

By applying the Pythagorean Theorem, we incorporate the newly found values and solve for the unknowns.
Recognizing the properties and structure of right triangles helps break down complex geometric problems into simpler, solvable equations through familiar principles.
Ultimately, understanding and utilizing the characteristics of right triangles is key to deriving precise measurements and ensuring our calculations align with the given conditions.