Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

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Solution

Median divides the opposite side.

∴

Given that,

In ΔABD and ΔPQM,

(Proved above)

∴ ΔABD ∼ ΔPQM (By SSS similarity criterion)

⇒ ∠ABD = ∠PQM (Corresponding angles of similar triangles)

In ΔABC and ΔPQR,

∠ABD = ∠PQM (Proved above)

∴ ΔABC ∼ ΔPQR (By SAS similarity criterion)

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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that

Δ A B C \sim Δ P Q R

Q. Question 3 (ii)
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:

Δ A B C ≅ Δ P Q R

Q. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that

Δ A B C \sim Δ P Q R

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that:

(i) ΔABM ≅ ΔPQN

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and

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and median

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Δ A B C = Δ P Q R

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Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of ΔPQR (see the given figure). Show that ΔABC ∼ ΔPQR.

Median divides the opposite side. ∴ Given that, In ΔABD and ΔPQM, (Proved above) ∴ ΔABD ∼ ΔPQM (By SSS similarity criterion) ⇒ ∠ABD = ∠PQM (Corresponding angles of similar triangles) In ΔABC and ΔPQR, ∠ABD = ∠PQM (Proved above) ∴ ΔABC ∼ ΔPQR (By SAS similarity criterion)