Question

P, and Q are the midpoints of AB & AC in triangleABC. If area of triangle APQ = 12√3, find the area of triangleABC​

Answer 1

Answer:

Since P and Q are midpoints of AB and AC ,

we have ,

AP/AB = AQ/AC = PQ/BC = 1:2

Now, we know that area ratio = (base ratio)²

so, Area ∆APQ : Area ∆ ABC = PQ² : BC² = 1:4

Hence ,

1-------------12√3

4------------48√3 (Ans)

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Answer 2

The area of triangle ABC=$48\sqrt3$ square units

Step-by-step explanation:

P and Q are mid-point of AB and AC

AB=2 PQ

AC=2 AQ

Area of triangle APQ=$12\sqrt 3$ square units

By mid-point segment theorem

PQ is parallel to BC

Angle APQ=Angle ABC( Corresponding angles are equal)

Angle PAQ=Angle BAC (Common angle)

Triangle ABC is similar to triangle APQ by AA similarity postulate.

When two triangles are similar

Then , the ratio of their areas=Ratio of square of their corresponding sides

Therefore,

$\frac{ar(ABC)}{ar(APQ)}=(\frac{AB}{AP})^2$

Substitute the values

$\frac{ar(ABC)}{12\sqrt 3}=(\frac{2AP}{AP})^2=4$

$ar(ABC)=48\sqrt3$ square units

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