Question

In the diagram, DEF MNP. Find the value of x. And m<N

De = 2
Df = x - 4/3
Angles are 89 and 40

Mn = 3x
Mp = x^2 + 16

Answer 1

To solve for \( x \) given the information about the triangles DEF and MNP where \( \triangle DEF \sim \triangle MNP \), we can use the properties of similar triangles, which state that corresponding sides are proportional and corresponding angles are congruent.

1. **Set up the proportions:**
From the similarity, we have:
\[
\frac{DE}{MN} = \frac{DF}{MP} = \frac{EF}{NP}
\]

Plugging in the given values:
- \( DE = 2 \)
- \( DF = \frac{x - 4}{3} \)
- \( MN = 3x \)
- \( MP = x^2 + 16 \)

We can compare \( DE \) and \( MN \):
\[
\frac{2}{3x} = \frac{\frac{x - 4}{3}}{x^2 + 16}
\]

2. **Cross-multiply to eliminate the fractions:**
\[
2(x^2 + 16) = (3x)(x - 4)
\]

Expanding both sides gives:
\[
2x^2 + 32 = 3x^2 - 12x
\]

3. **Rearranging the equation:**
Collect all terms on one side:
\[
0 = 3x^2 - 12x - 2x^2 - 32
\]
\[
0 = x^2 - 12x - 32
\]

This can be rewritten as:
\[
x^2 - 12x - 32 = 0
\]

4. **Use the quadratic formula to solve for \( x \):**
The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Here, \( a = 1, b = -12, c = -32 \):
\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(-32)}}{2(1)}
\]
\[
x = \frac{12 \pm \sqrt{144 + 128}}{2}
\]
\[
x = \frac{12 \pm \sqrt{272}}{2}
\]
\[
x = \frac{12 \pm 16.49}{2}
\]

**Calculate values:**
\[
x = \frac{12 + 16.49}{2} \text{ and } \frac{12 - 16.49}{2}
\]
\[
x_1 \approx \frac{28.49}{2} \approx 14.245
\]
\[
x_2 \approx \frac{-4.49}{2} \approx -2.245
\]

Since \( x \) is a length, we reject the negative value. Therefore:
\[
x \approx 14.245
\]

5. **Conclusion: Relate angle comparison \( m < N \):**
Since \( \triangle DEF \) and \( \triangle MNP \) are similar and you have mentioned angles of 89 and 40 degrees, angle comparisons will depend on the triangle correspondence; however, if the angles are arranged properly, then the condition \( m < N \) should hold true based on that relationship.

Thus, the final solution is:
\[
x \approx 14.245
\]