Question

In the figure, M is the
midpoint of PO and
MN | QR
If MN = 5 cm
PQ = 12 cm, PR = 13
cm, then the perimeter
of APQR is ....​

Answer 1

Given:

M is the midpoint of PO and MN // QR

MN = 5 cm

PQ = 12 cm

PR = 13 cm

To find:

The perimeter of Δ PQR

Solution:

We know,

$\boxed{\bold{Midpoint \:Theorem}}:$ A line segment in a triangle joining the midpoint of two sides of the triangle is parallel to its third side and is also half of the length of the third side.

$\boxed{\bold{Converse\: Midpoint \:Theorem}}:$ If a line segment is drawn through the midpoint of any one side of a triangle and parallel to the other side, then the line segment bisects the third side of the triangle.

Here, we have

MN // QR and M is a midpoint of PQ, according to the Converse Midpoint theorem,

∴ N is a midpoint of PR

So, according to the midpoint theorem, we get

$QR = 2 \times MN = 2 \times 5 = 10 \:cm$

Now,

The perimeter of Δ PQR,

= sum of all 3 sides

= PQ + PR + QR

= 12 + 13 + 10

= 35 cm

Thus, the perimeter of the Δ PQR is → 35 cm.

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