in triangle abc, P,Q,R are mid points. Area of triangle PQR is equal to 16 cm square then find the area of ABC
Answers
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Answer:
The area of triangle ABC is 64 cm².
Step-by-step explanation:
In triangle ABC, P,Q,R are mid points. Area of triangle PQR is equal to 16 cm squared, then find the area of ABC.
By drawing the figure based on the given, we can apply the Midpoint Theorem of the triangle which states that the line segment that connects the midpoints of two sides of the triangle is parallel and congruent to one-half of the third side.
Let us prove that ΔPQR ≅ ΔCRQ.
Statement Reason
1. PR ║ QC 1. Midpoint Theorem of the Triangle
2. PR ≅ QC 2. Midpoint Theorem of the Triangle
3. ∠PRQ ≅ ∠CQR 3. Alternate interior angles are congruent if two parallel
lines are cut by transversal
4. QR ≅ QR 4. Reflexive Property
5. ΔPQR ≅ ΔCRQ 5. SAS Congruence
Let us prove that ΔPQR ≅ ΔQPA.
Statement Reason
1. PR ║ AQ 1. Midpoint Theorem of the Triangle
2. PR ≅ AQ 2. Midpoint Theorem of the Triangle
3. ∠QPR ≅ ∠PQA 3. Alternate interior angles are congruent if two parallel
lines are cut by transversal
4. PQ ≅ PQ 4. Reflexive Property
5. ΔPQR ≅ ΔQPA 5. SAS Congruence
Let us prove that ΔPQR ≅ ΔRBP.
Statement Reason
1. PQ ║ BR 1. Midpoint Theorem of the Triangle
2. PQ ≅ BR 2. Midpoint Theorem of the Triangle
3. ∠QPR ≅ ∠BRQ 3. Alternate interior angles are congruent if two parallel
lines are cut by transversal
4. PR ≅ PR 4. Reflexive Property
5. ΔPQR ≅ ΔRBP 5. SAS Congruence
Thus,
ΔPQR ≅ ΔCRQ ≅ ΔQPA = ΔRBP
Area of ΔPQR = 16 cm²
Area of ΔCRQ = 16 cm²
Area of ΔQPA = 16 cm²
Area of ΔRBP = 16 cm²
Area of ΔABC = Area of ΔPQR + Area of ΔCRQ + Area of ΔQPA + Area of ΔRBP
Area of ΔABC = 16 cm² + 16 cm² + 16 cm² + 16 cm²
Area of ΔABC = 64 cm²