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If the answer of point 3 is 35,3 and the answer of point 4 is 9,4 please give me the process i need to follow to reach that answers

To solve for \(x\) in each triangle (points 3 and 4), we'll apply the Pythagorean Theorem, which is used to find the length of a side in a right triangle. The theorem states:

\[
c^2 = a^2 + b^2
\]

where \(c\) is the hypotenuse (the side opposite the right angle), and \(a\) and \(b\) are the other two sides.

### **Point 3:**
Given:
- \(a = 31\)
- \(b = x\)
- \(c = 48\)

Apply the Pythagorean Theorem:

\[
48^2 = 31^2 + x^2
\]

Calculate the squares:

\[
2304 = 961 + x^2
\]

Subtract 961 from both sides:

\[
2304 - 961 = x^2
\]

\[
1343 = x^2
\]

Take the square root of both sides:

\[
x = \sqrt{1343} \approx 36.65
\]

This doesn't match your expected result, so let's double-check the given numbers. However, if the answer was given as 35.3, it might be an approximation or a different method was applied, but the calculation is close.

### **Point 4:**
Given:
- \(a = 3x\)
- \(b = x\)
- \(c = 20\)

Apply the Pythagorean Theorem:

\[
20^2 = (3x)^2 + x^2
\]

Calculate:

\[
400 = 9x^2 + x^2
\]

Combine like terms:

\[
400 = 10x^2
\]

Divide by 10:

\[
x^2 = 40
\]

Take the square root of both sides:

\[
x = \sqrt{40} \approx 6.32
\]

If the answer was 9.4, let's adjust to match:

Given that the expected value was 9.4, it's possible there was an arithmetic error. Using the direct calculation, the square root of 40 should have been:

\[
x \approx 6.32
\]

Let me clarify if this matches your understanding or there was a typo in the question.

---

Would you like any further details or do you have any questions? Here are five related questions you might find interesting:

1. How can we use the cosine rule when a triangle isn't right-angled?
2. What are the steps to solve a triangle using the sine rule?
3. How does the area formula for triangles change when using Heron's formula?
4. Can the Pythagorean theorem be used in non-Euclidean geometries?
5. What are some practical applications of the Pythagorean theorem in real life?

**Tip:** Always double-check whether a triangle is right-angled before applying the Pythagorean theorem—this is a common source of error!

If the answer of point 3 is 35.3 and the answer of point 4 is 9.4, please give me the process I need to follow to reach those answers.

Learn how to solve right triangle problems using the Pythagorean Theorem. This example covers two problems, where the answers for x are 35.3 and 9.4, respectively. Suitable for students in Grades 9-10.

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