Step 1:

To determine the value of a in the given right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Step 2:

In triangle WYZ:

\(WZ^ 2= WY^ 2+ YZ^ 2= 4^ 2+ 3^ 2= 25\)

\(WZ= 5\)

Step 3:

In triangle WYX:

\(WX^ 2= WY^ 2+ YX^ 2\)

\(b^ 2= a^ 2+ 4^ 2= a^ 2+ 16\)

Step 4:

In triangle WXZ:

\(XZ^ 2= WX^ 2+ WZ^ 2\)

\(( a+ 3) ^ 2= b^ 2+ 5^ 2\)

\(a^ 2+ 6a+ 9= b^ 2+ 25\)

Step 5:

Calculation:z

\(a^ 2+ 6a+ 9= a^ 2+ 16+ 25\)

\(6a= 16+ 25- 9\)

\(6a= 32\)

\(a= \frac { 32} { 6} = \frac { 16} { 3} \)

Supplemental Knowledge:

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. 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. The formula is:

\(a^ 2+ b^ 2= c^ 2\)

This theorem can be used to find missing side lengths in right triangles when two sides are known.

From Concepts to Reality:

Imagine designing an accessible ramp and need to meet certain specifications; using the Pythagorean Theorem helps determine its length and angle to ensure safety and compliance.

Applying this theorem allows you to calculate how long a ramp needs to be in order for its slope to meet regulatory guidelines.

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