In Δ ABC, ∠B=35°, ∠C=65° and the bisector of ∠ABC meets BC in P. Arrange AP, BP and CP in descending order.
Answers
BP > CP > AP
Step-by-step explanation:
In triangle Δ ABC, ∠ B = 35°, ∠ C = 65° and the bisector of ∠ B meets AC at P.
Now, ∠ A = 180° - 35° - 65° = 80°
So, in triangle Δ ABC, ∠ A > ∠ C, hence, BC > AB
Now, line BP bisects ∠ B and intersects the line AC at P.
So, {As given above}
Then, CP > AP
Now, it is clear that, BP > CP > AP {In descending order} (Answer)
Answer:
Therefore, BP > AP > CP is the descending order.
Step-by-step explanation:
✯Question:-
In △ ABC, ∠ B = 35°, ∠ C = 65° and the bisector of ∠ BAC meets BC in P. Arrange AP, BP and CP in descending order.
✯Solution:-
Consider △ ABC
By sum property of a triangle
∠ A + ∠ B + ∠ C = 180°
To find ∠ A
∠ A = 180° – ∠ B – ∠ C
By substituting the values
∠ A = 180° – 35° – 65°
By subtraction
∠ A = 180° – 100°
∠ A = 80°
We know that
∠ BAP = ½ ∠ A
So we get
∠ BAP = ½ (80°)
By division
∠ BAP = 40°
Consider △ ABP
It is given that ∠ B = 35° and
∠ BAP = 40°
By sum property of a triangle
∠ BAP + ∠ BPA + ∠ PBA = 180°
To find ∠ BPA
∠ BPA = 180° – ∠ BAP – ∠ PBA
By substituting values
∠ BPA = 180° – 35° – 40°
By subtraction
∠ BPA = 180° – 75°
∠ BPA = 105°
We know that ∠ B is the smallest angle and the side opposite to it i.e. AP is the smallest side.
So we get AP < BP ….. (1)
Consider △ APC ∠ CAP = ½ ∠ A
So we get
∠ CAP = ½ (80°)
By division
∠ CAP = 40°
By sum property of a triangle
∠ APC + ∠ CAP + ∠ CPA = 180°
To find ∠ APC
∠ APC = 180° – ∠ CAP – ∠ CPA
By substituting values
∠ APC = 180° – 40° – 65°
So we get
∠ APC = 180° – 105°
By subtraction
∠ APC = 75°
So we know that ∠ CAP is the smallest angle and the side opposite to it i.e. CP is the smallest side.
We get CP < AP …… (2)
By considering equation (1) and (2)
BP > AP > CP
Therefore, BP > AP > CP is the descending order.
✯Answer:-
Therefore, BP > AP > CP is the descending order.
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