vonsider the isosceles trapezoid pqrs. the bases are pq which is equal to 120mm, rs is equal to 62 mm and the arm is equal to 48 mm. find the height of the trapezoid, diagonal length, and the area of the trapezoid

Let's break down the problem step by step.

### 1. **Height of the Isosceles Trapezoid:**
The isosceles trapezoid has two parallel bases, \( PQ \) and \( RS \), with \( PQ = 120 \, \text{mm} \) and \( RS = 62 \, \text{mm} \), and each non-parallel side (arm) \( PS = QR = 48 \, \text{mm} \).

To find the height (\( h \)) of the trapezoid, we use the following approach:
- The difference between the lengths of the two bases is \( PQ - RS = 120 - 62 = 58 \, \text{mm} \).
- Since the trapezoid is isosceles, the legs are symmetric, meaning half of this difference is distributed equally on both sides: \( \frac{58}{2} = 29 \, \text{mm} \).

Now, using the Pythagorean theorem in the right triangle formed by the height \( h \), half the difference between the bases (29 mm), and the arm \( PS = 48 \, \text{mm} \):

\[
h^2 + 29^2 = 48^2
\]
\[
h^2 + 841 = 2304
\]
\[
h^2 = 2304 - 841 = 1463
\]
\[
h = \sqrt{1463} \approx 38.24 \, \text{mm}
\]

### 2. **Diagonal Length:**
Next, to find the length of the diagonal (\( PR \) or \( QS \)), we use the Pythagorean theorem again in the right triangle formed by the height \( h \), half the sum of the bases \( \frac{PQ + RS}{2} = \frac{120 + 62}{2} = 91 \, \text{mm} \), and the diagonal \( d \):

\[
d^2 = h^2 + \left(\frac{PQ + RS}{2}\right)^2
\]
\[
d^2 = 1463 + 91^2
\]
\[
d^2 = 1463 + 8281 = 9744
\]
\[
d = \sqrt{9744} \approx 98.71 \, \text{mm}
\]

### 3. **Area of the Trapezoid:**
The area \( A \) of a trapezoid is given by the formula:

\[
A = \frac{1}{2} \times (PQ + RS) \times h
\]
\[
A = \frac{1}{2} \times (120 + 62) \times 38.24
\]
\[
A = \frac{1}{2} \times 182 \times 38.24 = 91 \times 38.24 \approx 3479.84 \, \text{mm}^2
\]

### Final Results:
- **Height** of the trapezoid: \( \approx 38.24 \, \text{mm} \)
- **Diagonal length**: \( \approx 98.71 \, \text{mm} \)
- **Area** of the trapezoid: \( \approx 3479.84 \, \text{mm}^2 \)

---

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:
1. How does the Pythagorean theorem apply in solving geometric problems like this?
2. How would changing the length of the arms affect the height of the trapezoid?
3. What is the formula for the area of a trapezoid and how does it relate to that of a parallelogram?
4. How can the height of a non-isosceles trapezoid be found?
5. How can trigonometry be used to find the diagonal in a trapezoid?

**Tip**: When working with geometric shapes, sketching the figure and identifying right triangles can help simplify complex problems.

Consider the isosceles trapezoid PQRS. The bases are PQ which is equal to 120mm, RS is equal to 62mm, and the arm is equal to 48mm. Find the height of the trapezoid, diagonal length, and the area of the trapezoid.

Solve for the height, diagonal length, and area of an isosceles trapezoid with bases PQ = 120mm, RS = 62mm, and arm length 48mm. Learn how to apply the Pythagorean theorem and area formula step-by-step. Suitable for Grades 9-11 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation