the base of the triangle is not 13cm and the height is not 21cm. you need to use pythagorus theorum to find the height

You're correct. The values given (13 cm and 21 cm) actually refer to the sides of the triangle, not the base and height. Let's go through the correct steps using the Pythagorean theorem to find the height of the triangle and then calculate the surface area again.

### Step 1: Identify the sides of the triangle
- One side: 13 cm
- Another side (base): 21 cm
- Hypotenuse: 35 cm

### Step 2: Find the height of the triangle using the Pythagorean theorem
The height of the triangle can be found by applying the Pythagorean theorem in the context of the right triangle formed by the height, the base (21 cm), and the hypotenuse (35 cm).

Let's denote the height as \( h \) and the base segment perpendicular to the height as \( b_1 \).

We know:

\[
\text{Hypotenuse}^2 = \text{Height}^2 + \text{Base Segment}^2
\]

Since the triangle sides are 13 cm and 21 cm, and the hypotenuse is 35 cm, we find \( h \) and \( b_1 \).

### Step 3: Calculating the height
Given:
\[
\text{Hypotenuse}^2 = \text{Height}^2 + \text{Base Segment}^2
\]

The area of a triangle is also given by:

\[
\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]

The relationship between the sides and the height:

Using Pythagoras:

\[
h^2 = 35^2 - 21^2
\]

\[
h = \sqrt{1225 - 441} = \sqrt{784} = 28 \text{ cm}
\]

So, the height is 28 cm. Let's check using the relationship:

### Step 4: Calculate the area of the triangular faces

\[
\text{Area of each triangle} = \frac{1}{2} \times 21 \, \text{cm} \times 28 \, \text{cm} = 294 \, \text{cm}^2
\]

There are two triangular faces:

\[
\text{Total area of the triangular faces} = 2 \times 294 = 588 \, \text{cm}^2
\]

### Step 5: Area of the rectangular faces

1. **Face 1** (connected to the 21 cm side):
   \[
   \text{Area} = 21 \times 35 = 735 \, \text{cm}^2
   \]
2. **Face 2** (connected to the 13 cm side):
   \[
   \text{Area} = 13 \times 35 = 455 \, \text{cm}^2
   \]
3. **Face 3** (connected to the hypotenuse side, 35 cm):
   \[
   \text{Area} = 35 \times 35 = 1225 \, \text{cm}^2
   \]

### Step 6: Total Surface Area

\[
\text{Total Surface Area} = 588 + 735 + 455 + 1225 = 3003 \, \text{cm}^2
\]

### Step 7: Round to 3 Significant Figures

\[
\text{Surface Area} = 3000 \, \text{cm}^2
\]

The surface area of the solid is **3000 cm²**.

Would you like further details or have any questions?

Here are five related questions for further practice:
1. How does the angle of the triangular prism affect its surface area?
2. What would happen to the surface area if one of the triangular sides was reduced by 25%?
3. How can we calculate the volume of this triangular prism?
4. If the hypotenuse were shortened, how would the height and surface area change?
5. How does the total surface area change if the length of the prism increases?

**Tip:** For complex problems, always start by carefully analyzing the figure and identifying all given dimensions before proceeding with calculations.

Learn how to apply the Pythagorean theorem to find the height of a triangle and calculate the surface area of a triangular prism. This problem involves geometry concepts, including the Pythagorean theorem, and surface area calculations for triangular prisms. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation