Find a missing side length when given perimeter (practice) | Khan ... - Free Printable

Problem Analysis:

The given image depicts a trapezoid with the following dimensions:
- The top base (shorter parallel side) is 6 meters.
- The bottom base (longer parallel side) is 10 meters.
- One of the non-parallel sides (slant height) is 8 meters.
- The height of the trapezoid (the perpendicular distance between the two parallel sides) is unknown and needs to be determined.

We are tasked with finding the height of the trapezoid.

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Step-by-Step Solution:

#### Step 1: Understand the Geometry
A trapezoid has one pair of parallel sides. In this case, the top base is 6 meters, and the bottom base is 10 meters. The difference in length between the two bases is:
$$
10 - 6 = 4 \text{ meters}
$$
This difference is distributed equally on both sides of the trapezoid when we drop perpendiculars from the endpoints of the top base to the bottom base. Therefore, each segment on the bottom base that extends beyond the top base is:
$$
\frac{4}{2} = 2 \text{ meters}
$$

#### Step 2: Form a Right Triangle
By dropping a perpendicular from one endpoint of the top base to the bottom base, we create a right triangle. The components of this right triangle are:
- One leg is the height of the trapezoid, which we need to find.
- The other leg is the horizontal distance from the foot of the perpendicular to the endpoint of the top base, which is 2 meters (as calculated above).
- The hypotenuse is the slant height of the trapezoid, which is given as 8 meters.

#### Step 3: Apply the Pythagorean Theorem
The Pythagorean theorem states:
$$
\text{hypotenuse}^2 = \text{leg}_1^2 + \text{leg}_2^2
$$
Here:
- The hypotenuse is 8 meters.
- One leg is the height of the trapezoid, which we denote as \( h \).
- The other leg is 2 meters.

Substitute the known values into the Pythagorean theorem:
$$
8^2 = h^2 + 2^2
$$
$$
64 = h^2 + 4
$$

#### Step 4: Solve for \( h \)
Rearrange the equation to isolate \( h^2 \):
$$
h^2 = 64 - 4
$$
$$
h^2 = 60
$$
Take the square root of both sides:
$$
h = \sqrt{60}
$$
Simplify the square root:
$$
h = \sqrt{4 \times 15} = 2\sqrt{15}
$$

#### Step 5: Final Answer
The height of the trapezoid is:
$$
\boxed{2\sqrt{15}}
$$

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Explanation:

1. We used the properties of trapezoids to identify the right triangle formed by dropping a perpendicular from the top base to the bottom base.
2. The Pythagorean theorem was applied to relate the height, the horizontal distance, and the slant height.
3. Simplification of the square root provided the exact value of the height.

Thus, the height of the trapezoid is \( \boxed{2\sqrt{15}} \) meters.