Question

Points A and B are midpoints of the sides of triangle QRS.

Triangle Q R S is cut by line segment A B. Point A is the midpoint of side Q S and point B is the midpoint of side R S. The length of Q R is 6 meters, the length of Q A is 4 meters, the length of A B is 3 meters, and the length of R B is 3 meters.

What is SA?

a 2 m
b 3 m
c 4 m
d 6 m

Answer 1

Given:

QRS is a triangle

A is the midpoints of side QS

B is the midpoint of side RS

QR = 6 m

QA = 4 m

AB = 3 m

RB = 3 m

To find:

The length of SA

Solution:  

The given problem can be solved by two methods which are as follows:

METHOD 1:

The length of QA = 4 m

A is the midpoint of side QS of Δ QRS

∴ QA = SA

⇒ SA = 4 m ← option (c)

Thus, the length of SA is 4 m.

METHOD 2:

We know that

The Midpoint Theorem states that the line joining the midpoints of two sides of a triangle is parallel to the third side.

∴ AB // QR

Now, considering ΔABS and ΔQRS, we have

∠S = ∠S ..... [common angle]

∠SAB = ∠SQR ...... [corresponding angles]

∴ ΔABS ~ ΔQRS ...... [By AA similarity]

Since the corresponding sides of two similar triangles are proportional to each other.

∴ $\frac{AB}{QR} = \frac{SA}{QS}$

⇒ $\frac{AB}{QR} = \frac{SA}{QA\:+\:SA}$

we will substitute the given values of AB, QR & QA to find SA

⇒ $\frac{3}{6} = \frac{SA}{4\:+\:SA}$

⇒ $\frac{1}{2} = \frac{SA}{4\:+\:SA}$

⇒ $4 + SA = 2SA$

⇒ $4 = 2SA - SA$

⇒ SA = 4 m

Thus, the length of SA is 4 meters.

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