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Construction
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11.6
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:
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To construct a triangle, given its perimeter and its two base angles.

Given the base angles, ∠B and ∠C and AB + AC + BC, we have to construct ΔABC.

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Steps of Construction
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:

- Draw a line segment XY equal to AB + AC + BC.
- Make ∠ LXY equal to ∠B and ∠ MYX equal to ∠C.
- Bisect ∠ LXY and ∠ MYX. Let these bisectors intersect at a point A .
- Draw perpendicular bisectors of line AX and AY
- Let PQ intersect XY at B and RS intersect XY at C.
- Join AB and AC

Thus, Δ ABC is a the required triangle

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Justification
**

To justify, we have to prove

- AB + BC + AC = XY
- ∠ LXY = ∠ B
- ∠ MYX = ∠ C

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In
**
**
Δ
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**
AXQ
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,

PQ is the perpendicular bisector of AX,

∴ BX = BA

Similarly, we can say

CY = CA

Now, we know that

XY =
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XQ
**
+ BC +
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CY
**

XY =
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AB
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+ BC +
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AC
**

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In
**
**
Δ
**
**
AXQ
**

Since AX = BA

∴ ∠ BAX = ∠ AXB

Now, ∠ ABC is the exterior angle of triangle AXQ

∠ ABC = ∠ BAX + ∠ AXB

∠ ABC = ∠ AXB + ∠ AXB

∠ ABC = 2 ∠ AXB

∠ ABC = ∠ LXY

Thus, ∠ B = ∠ LXY

Similarly, we can prove ∠ A = ∠ MYX

Hence justified