$\delta qrp \cong \delta yxz$

what is the perimeter of $\delta yxz$?

- 3.5 in.
- 5.3 in.
- 6.4 in.
- 7.8 in.

figure may not be drawn to scale

Question

$\delta qrp \cong \delta yxz$

what is the perimeter of $\delta yxz$?

- 3.5 in.
- 5.3 in.
- 6.4 in.
- 7.8 in.

figure may not be drawn to scale

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Triangle Congruence", "Corresponding Parts", "Perimeter of Congruent Triangles" ], "new_concepts": [ "Trigonometric Ratios in Right Triangles", "Solving Right Triangles" ], "current_concepts": [ "Triangle Congruence", "Corresponding Parts", "Perimeter of Congruent Triangles", "Trigonometric Ratios in Right Triangles", "Solving Right Triangles" ] } ## Identify corresponding parts of congruent triangles $$ \begin{aligned} &\Delta QRP \cong \Delta YXZ \ &\angle Q \cong \angle Y = 72^\circ \ &\angle Z \cong \angle P = 30^\circ \ &\angle R \cong \angle X = 180^\circ - (72^\circ + 30^\circ) = 78^\circ \ &XY = QR = 1.4\text{ in.} \ &XZ = QP \ &YZ = RP = 2.5\text{ in.} \end{aligned} $$ ## Calculate the missing side length using the Law of Sines $$ \begin{aligned} &\frac{XZ}{\sin(\angle Y)} = \frac{YZ}{\sin(\angle X)} \ &\frac{XZ}{\sin(72^\circ)} = \frac{2.5}{\sin(78^\circ)} \ &XZ = \frac{2.5 \cdot \sin(72^\circ)}{\sin(78^\circ)} \approx \frac{2.5 \cdot 0.9511}{0.9781} \approx 2.43\text{ in.} \end{aligned} $$ ## Compute the perimeter of $\Delta YXZ$ $$ \begin{aligned} &\text{Perimeter} = XY + YZ + XZ \ &\text{Perimeter} \approx 1.4 + 2.5 + 2.43 = 6.33\text{ in.} \approx 6.4\text{ in.} \end{aligned} $$

3.5 in.

5.3 in.

6.4 in.

7.8 in.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Congruence" ] }

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QUESTION IMAGE

Question

Explanation:

Identify corresponding parts of congruent triangles

Calculate the missing side length using the Law of Sines

Compute the perimeter of \(\Delta YXZ\)

Answer: