https://file.aiquickdraw.com/mathbot/file/17264250052578uvxiqg4.jpg

The triangle shown is a right triangle with a base of 10 units, a height of \( 2x \), and a hypotenuse that is split into two segments: \( 3x \) and \( x + 6 \).

To find the area of the triangle, we can use the formula:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

The base is given as 10, and the height is \( 2x \). So, the area formula becomes:

\[
\text{Area} = \frac{1}{2} \times 10 \times 2x = 10x
\]

Now, we need to find \( x \). To do this, we can use the Pythagorean theorem because this is a right triangle. The Pythagorean theorem states that:

\[
(\text{base})^2 + (\text{height})^2 = (\text{hypotenuse})^2
\]

Substituting the given lengths, we have:

\[
(3x)^2 + (2x)^2 = (x + 6)^2
\]

Now let's solve this equation:

\[
9x^2 + 4x^2 = (x + 6)^2
\]
\[
13x^2 = x^2 + 12x + 36
\]
\[
13x^2 - x^2 - 12x - 36 = 0
\]
\[
12x^2 - 12x - 36 = 0
\]

Now, divide the whole equation by 12:

\[
x^2 - x - 3 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-3)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 12}}{2}
\]
\[
x = \frac{1 \pm \sqrt{13}}{2}
\]

Therefore, \( x = \frac{1 + \sqrt{13}}{2} \) or \( x = \frac{1 - \sqrt{13}}{2} \).

Since \( x \) must be positive, we take the positive root:

\[
x = \frac{1 + \sqrt{13}}{2}
\]

Now, substitute this value of \( x \) back into the area formula \( \text{Area} = 10x \) to find the area.

Would you like me to calculate this area for you?

Find the area of the triangle with sides 3x, 2x, and x + 6, with a base of 10 units.

Learn how to find the area of a right triangle with sides 3x, 2x, and x + 6, and a base of 10 units. This detailed solution involves the Pythagorean theorem, solving a quadratic equation, and applying the area formula. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation