Example 3.41 from Elementary Algebra, 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem: example 3.41

A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x

Kelvin is building a gazebo and wants to brace each corner by placing a $10 ″$ piece of wood diagonally as shown above.

If he fastens the wood so that the ends of the brace are the same distance from the corner, what is the length of the legs of the right triangle formed? Approximate to the nearest tenth of an inch.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. How far up the wall does the ladder reach?

A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.

How did it go?

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let $x =$ the distance from the corner.
Step 4. Translate Write the appropriate formula and substitute.	$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ x^{2} + x^{2} & = & 10^{2} \end{matrix}$
Step 5. Solve the equation. Isolate the variable. Use the definition of square root. Simplify. Approximate to the nearest tenth.	$\begin{matrix} 2 x^{2} & = & 100 \\ x^{2} & = & 50 \\ x & = & \sqrt{50} \\ x & \approx & 7.1 \end{matrix}$
Step 6. Check. $\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ {(7.1)}^{2} + {(7.1)}^{2} & \approx & 10^{2} Yes. \end{matrix}$
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let $x =$ the distance from the corner.
Step 4. Translate Write the appropriate formula and substitute.	$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ x^{2} + x^{2} & = & 10^{2} \end{matrix}$
Step 5. Solve the equation. Isolate the variable. Use the definition of square root. Simplify. Approximate to the nearest tenth.	$\begin{matrix} 2 x^{2} & = & 100 \\ x^{2} & = & 50 \\ x & = & \sqrt{50} \\ x & \approx & 7.1 \end{matrix}$
Step 6. Check. $\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ {(7.1)}^{2} + {(7.1)}^{2} & \approx & 10^{2} Yes. \end{matrix}$
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem: example 3.41

Solution

Your turn

Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem: example 3.41

Solution

Your turn