Question: 19. Consider the isosceles triangle LMN and a student’s work to find the value of $x$.
$82^\circ$
$(9x + 4)^\circ$
$(12x - 11)^\circ$
Student's Work
Step | Work
1 | $9x + 4 = 12x - 11$
2 | $-3x = -15$
3 | $x = 5$
Which statement is true?
A The student makes a mistake in Step 1 because the student should have written $x + 4 = 82$.
B The student makes a mistake in Step 1 because the student should have written $x + 4 + 82 = 180$.
C The student makes a mistake in Step 3 because the student should have written $x = -12$.
D The student solves for $x$ correctly.
1. **Problem Statement:** We have an isosceles triangle LMN with angles $82^\circ$ at vertex M, $(9x + 4)^\circ$ at vertex L, and $(12x - 11)^\circ$ at vertex N. The student tries to find $x$ by setting $9x + 4 = 12x - 11$.
2. **Important Rule:** In any triangle, the sum of the interior angles is always $180^\circ$.
3. **Isosceles Triangle Property:** Two sides are equal, so their opposite angles are equal. Since the triangle is isosceles, the two base angles are equal. Here, the base angles are at L and N, so:
$$9x + 4 = 12x - 11$$
4. **Check Student's Step 1:** The student sets $9x + 4 = 12x - 11$ to find $x$. This assumes the base angles are equal, which is correct for an isosceles triangle.
5. **Solve the equation:**
$$9x + 4 = 12x - 11$$
Subtract $9x$ from both sides:
$$\cancel{9x} + 4 = 12x - 11 - \cancel{9x}$$
$$4 = 3x - 11$$
Add $11$ to both sides:
$$4 + 11 = 3x - 11 + 11$$
$$15 = 3x$$
Divide both sides by $3$:
$$\frac{15}{\cancel{3}} = \frac{3x}{\cancel{3}}$$
$$5 = x$$
6. **Check the sum of angles:**
Calculate each angle:
- Angle at L: $9(5) + 4 = 45 + 4 = 49^\circ$
- Angle at N: $12(5) - 11 = 60 - 11 = 49^\circ$
- Angle at M: $82^\circ$
Sum:
$$49 + 49 + 82 = 180^\circ$$
7. **Conclusion:** The student's work is correct. The base angles are equal, the equation is set correctly, and the solution $x=5$ satisfies the triangle angle sum.
**Answer:** D The student solves for $x$ correctly.